A theory of coupled strain-gradient plasticity with species transport at small deformations

Abstract

This work presents a mechanical isotropic rate-independent theory for plastically deformed materials coupled with species transport. The mass and virtual power balances are natural ingredients used in this work to obtain appropriate local balance laws for species transport, macroscopic, and microscopic forces. The second law of thermodynamics is another key tool used to obtain thermodynamically consistent constitutive relations for the species flux and microscopic stresses. The free energy is approximated as a quadratic form and used to obtain the energetic microscopic stresses as linear combinations of their respective energy conjugates and the species densities. Rate-independent Mises flow rule is deduced in terms of accumulated plastic strain and species density. Furthermore, variational formulation for the coupled theory is obtained as a variational inequality.

1. Introduction

Many species, such as hydrogen atoms, are gases at room temperature and so their storage could pose a serious challenge. For instance, hydrogen species transports can severely degrade the ductility of metals, and being the lightest known element in the periodic table, the permeations of hydrogen gases through thin walls cause damage to metallic structures used for storing these gases. It is therefore worthwhile to study the interactions between the transport of some species and elasto-plastic materials [1–3]. Studies based on molecular simulations and experimental measurements have provided insight into diffusion processes of hydrogen in solids at different length scales. For instance, enormous enhancement of diffusivity in body-centered cubic metals have been observed, where hydrogen absorption and diffusion give rise to local hydrogen accumulation with consequent potential defects [4].

Successful attempts have been made to couple species diffusion to elasticity in the classical papers by Larché and Cahn [5, 6]. A detailed description and review of this coupling is found in the work of Gurtin et al. [7] where special case of the substitutional alloy is also discussed. One of the most conceptually challenging features of this coupled theory is associated with the formulation of energy flow due to species transport. This issue has been addressed in the monograph of Gurtin et al. [7]. Among the first few works on interaction of hydrogen species with plastic deformation is the paper by Sofronis and McMeeking [8]. Their theory is based on the postulate of local equilibrium proposed by Oriani [9]. Further extensions to the work of Sofronis and McMeeking [8] were made by Krom et al. [10] to account for trapping sites of hydrogen atoms within plastic materials. This latter theory has been widely used among researchers working on this subject [11–13] partly because of its ability to provide insight into lattice-dilation and hydrogen-induced reduction of plastic flow resistance. However, the Oriani’s microstructural hypothesis on which the work of Krom et al. [10] is based has limitations in formulating a consistent continuum level coupled theory. Within the context of a continuum theory, Anand [14] developed a thermo-mechanical coupled theory for hydrogen transport and large viscoplastic deformation which does not explicitly account for trapping of hydrogen species and without consideration to the Oriani’s postulate. In the same spirit, Di Leo and Anand [15] modified the work of Anand [14] to account for trapping sites of hydrogen species. This latter work placed the concept of equilibrium between hydrogen residing in interstitial lattice site and hydrogen trapped at microstructural defects within the framework of the first and second laws of thermodynamics.

The classical theories of continuum mechanics cannot model coupled species transport and elasto-plastic deformations in the presence of intrinsic material length scales. This implies that these classical theories cannot apprehend size-dependent behaviors of materials. However, experimental results have shown that metallic materials exhibit size-effect phenomena and that strain gradients promote strain hardening [16–18]. The continuum theories developed to circumvent this shortcoming of the classical theories are known as strain-gradient theories [19–32].

It has been predicted and argued that size effects in materials may play a crucial role in aiding hydrogen diffusion [33, 34], although such interactions are yet to be fully assessed experimentally. However, attempts have been made toward the theoretical study of strain-gradient plasticity coupled with hydrogen species transports. For instance, Kristensen et al. [35] proposed a gradient-based framework for determining hydrogen assisted fractures in plastic materials using combination of stress-assisted diffusion of species and hydrogen-sensitive phase field fracture formulation. The phase field fracture method in modeling hydrogen interactions with solids is also used in the works of Martínez-Pañeda et al. [36], Anand et al. [2], and Wu et al. [37]. At the moment, sufficient studies have not been conducted in understanding hydrogen embrittlement and diffusion in elastic-plastically deforming solids under small and large deformations. Thus, precise cause of embrittlement is yet to be well understood, and this has engendered continuous research. Furthermore, most studies on this subject do not account for gradient of plastic distortion which is essential in predicting size-dependent behavior for plastic materials interacting with species. Hence, the need to further study the interactions of species transports (such as ionic, atomic, molecular, or chemical transports) with gradient materials with a view to providing possible cause of species embrittlement in solids. This work presents a continuum mechanical coupled theory for multiple species transport and strain-gradient plasticity compatible with the laws of thermodynamics. Natural deductions from this formulation are the scalar-chemistry plastic strain and vector-chemistry plastic strain gradients which play important roles in measuring plastic diffusivities in solids.

2. Basic kinematic relations

In small deformation theory of elasto-plastic materials, the displacement gradient $\nabla \mathbf{u}$ admits the additive decomposition:

$$\nabla \mathbf{u}={\mathbf{H}}^e+{\mathbf{H}}^p,$$

(1)

into elastic part ${\mathbf{H}}^e$ and plastic part ${\mathbf{H}}^p$ with $\mathrm{tr}{\mathbf{H}}^p=0$ , where $\mathbf{u}$ is the displacement vector at a point $\mathbf{X}$ of a region $B$ occupied by body under consideration and at time $t\ge 0$ . The quantity ${\mathbf{H}}^e$ is called the elastic distortion which measures stretch and rotation of the underlying material structure, in this case a lattice. The quantity ${\mathbf{H}}^p$ is called the plastic distortion which accounts for defects in the material structure arising from formation and motion of dislocations. The identity $\mathrm{tr}{\mathbf{H}}^p=0$ (i.e., trace of plastic distortion vanishes) is known as the constraint of plastic incompressibility requiring that no volumetric change is accompanied by plastic deformation, which is essentially true for most metals.

The total strain $\mathbf{E}$ also admits the additive decomposition:

$$\mathbf{E}={\mathbf{E}}^e+{\mathbf{E}}^p,\mathrm{with}\mathrm{tr}{\mathbf{E}}^p=0,$$

(2)

into elastic strain ${\mathbf{E}}^e$ and plastic strain ${\mathbf{E}}^p$ , where ${\mathbf{E}}^e$ and ${\mathbf{E}}^p$ are the symmetric parts of ${\mathbf{H}}^e$ and ${\mathbf{H}}^p$ , respectively. A theory of strain-gradient plasticity assumes plastic irrotationality, i.e., the skew part of ${\mathbf{H}}^p$ is ignored. Hence, by taking the time derivative of equation (1) and assuming plastic irrotationality, we have:

$$\nabla \stackrel{·}{\mathbf{u}}={\stackrel{·}{\mathbf{H}}}^e+{\stackrel{·}{\mathbf{E}}}^p,\mathrm{tr}{\stackrel{·}{\mathbf{E}}}^p=0.$$

(3)

The plastic flow direction ${\mathbf{N}}^p$ in the absence of higher-order dissipative microscopic stress is defined as:

$${\mathbf{N}}^p=\frac{{\stackrel{·}{\mathbf{E}}}^p}{|{\stackrel{·}{\mathbf{E}}}^p|}\mathrm{for}{\stackrel{·}{\mathbf{E}}}^p\ne 0.$$

(4)

The accumulation of plastic strain $e^p=e^p(\mathbf{X},t)$ is defined via the initial value problem:

$${\stackrel{·}{e}}^p=|{\stackrel{·}{\mathbf{E}}}^p|\mathrm{with}{e}^p(\mathbf{X},0)=0.$$

(5)

Whenever ${\stackrel{·}{\mathbf{E}}}^p\ne 0$ , then following equations (4) and (5), equation (3) can be written as:

$$\nabla \stackrel{·}{\mathbf{u}}={\stackrel{·}{\mathbf{H}}}^e+{\stackrel{·}{e}}^p{\mathbf{N}}^p,\mathrm{tr}{\mathbf{N}}^p=0.$$

(6)

The basic rate-like variables are $\stackrel{·}{\mathbf{u}}$ , ${\stackrel{·}{\mathbf{H}}}^e$ , and ${\stackrel{·}{e}}^p$ . The use of codirectionality hypothesis [7] exempts the flow direction as a kinematic variable.

The Burgers tensor $\mathbf{G}$ is defined as:

$$\mathbf{G}=\nabla \times {\mathbf{E}}^p,(G_{\mathrm{ij}}={ϵ}_{\mathrm{irs}}{E}_{\mathrm{js},r}^p).$$

(7)

The Burgers tensor rate $\stackrel{·}{\mathbf{G}}$ can be written in terms of the accumulated plastic strain rate and its gradient $\nabla {\stackrel{·}{e}}^p$ viz:

$$\stackrel{·}{\mathbf{G}}=(\nabla \times {\mathbf{N}}^p){\stackrel{·}{e}}^p+(\nabla {\stackrel{·}{e}}^p\times ){\mathbf{N}}^p.$$

(8)

Basic to the present work is the assumption that the Burgers tensor $\mathbf{G}$ has the kinematic constitutive form:

$$\mathbf{G}=\hat{\mathbf{G}}(\nabla e^p),$$

(9)

where $\nabla e^p$ is the gradient of the accumulated plastic strain. By choosing $e^p$ and $\nabla e^p$ as the independent variables, it follows from equations (8) and (9) that:

$$\nabla \times {\mathbf{N}}^p=0,\mathrm{and}\frac{\partial G_{\mathrm{ij}}}{\partial e_{,k}^p}={ϵ}_{\mathrm{iks}}{N}_{\mathrm{sj}}^p.$$

(10)

3. Macroscopic and microscopic force balances

The basic kinematic rate variables ${\stackrel{·}{\mathbf{H}}}^e$ , $\stackrel{·}{\mathbf{u}}$ , and ${\stackrel{·}{e}}^p$ are conjugated by force-like variables. A region $P$ of the body $B$ is acted upon by external forces which are body forces $\mathbf{b}=\mathbf{b}(\mathbf{X},t)$ on $P$ and macrotractions $\mathbf{t}(\mathbf{n})$ on the boundary $\partial P$ of $P$ with outward unit normal $\mathbf{n}$ on $\partial P$ . The body force $\mathbf{b}$ and microtraction $\mathbf{t}$ are both power-conjugate to the velocity $\stackrel{·}{\mathbf{u}}$ . Also, we assume that there is an external microtraction $\mathbf{K}(\mathbf{n})$ power-conjugate to ${\stackrel{·}{\mathbf{E}}}^p$ on $\partial P$ . The internal forces are expressed through the symmetric Cauchy stress $\mathbf{T}$ power-conjugate to the elastic distortion rate ${\stackrel{·}{\mathbf{H}}}^e$ , and the plastic stress ${\mathbf{T}}^p$ power-conjugate to the plastic strain rate ${\stackrel{·}{\mathbf{E}}}^p$ . Without loss of generality, we assumed ${\mathbf{T}}^p$ is symmetric and deviatoric.

In order to account for size effects via the Burgers tensor, we assume that there is a microscopic stress ${\mathbf{S}}^p$ power-conjugate to the Burgers tensor rate $\stackrel{·}{\mathbf{G}}$ . This choice is, however, more restrictive than the use of a general form in which a third-order microscopic stress tensor ${\mathbb{K}}^p$ is power-conjugate to the gradient of the plastic strain rate $\nabla {\stackrel{·}{\mathbf{E}}}^p$ . The use of the conjugate pair $({\mathbf{S}}^p,\stackrel{·}{\mathbf{G}})$ is known to be physically meaningful in the study of single crystal–gradient plasticity, where the Burgers tensor $\mathbf{G}$ is determined by distributions of geometrically necessary dislocation densities. On the contrary, gradient of plastic strain includes in its expression quantities unidentified with any known dislocation densities Borokinni and Fadodun [38].

Following the identified conjugate pairs, the power ${\mathcal{W}}_{\mathrm{ext}}(P)$ expended by the external forces on $P$ is:

$${\mathcal{W}}_{\mathrm{ext}}(P)={\int }_{\partial P}\mathbf{t}(\mathbf{n})·\stackrel{·}{\mathbf{u}}\mathrm{da}+{\int }_P\mathbf{b}·\stackrel{·}{\mathbf{u}}\mathrm{dv}+{\int }_{\partial P}\mathbf{K}(\mathbf{n}):{\stackrel{·}{\mathbf{E}}}^p\mathrm{da}.$$

(11)

Also, the power ${\mathcal{W}}_{\mathrm{int}}(P)$ expended within $P$ is given by:

$${\mathcal{W}}_{\mathrm{int}}(P)={\int }_P[\mathbf{T}:{\stackrel{·}{\mathbf{H}}}^e+{\mathbf{T}}^p:{\stackrel{·}{\mathbf{E}}}^p+{\mathbf{S}}^p:\stackrel{·}{\mathbf{G}}]\mathrm{dv}.$$

(12)

Following the works of Gurtin et al. [7], we assume the codirectionality hypothesis which states that:

$$\frac{{\mathbf{T}}_o-{\mathbf{T}}_{\mathrm{back}}}{|{\mathbf{T}}_o-{\mathbf{T}}_{\mathrm{back}}|}=\frac{{\stackrel{·}{\mathbf{E}}}^p}{|{\stackrel{·}{\mathbf{E}}}^p|}={\mathbf{N}}^p,$$

(13)

where ${\mathbf{T}}_o$ is the deviatoric part of $\mathbf{T}$ , and ${\mathbf{T}}_{\mathrm{back}}$ is the back stress associated with the energetic part of microscopic stresses ${\mathbf{T}}^p$ and ${\mathbf{S}}^p$ .

Define scalar microtraction $K(\mathbf{n})$ , scalar plastic microscopic stress ${\tau }^p$ , and vector microscopic stress ${\mathit{\xi }}^p$ by:

$$K(\mathbf{n})=\mathbf{K}(\mathbf{n}):{\mathbf{N}}^p,{\tau }^p={\mathbf{T}}^p:{\mathbf{N}}^p\mathrm{and}{\xi }_k^p=S_{\mathrm{ij}}^p{ϵ}_{\mathrm{iks}}{N}_{\mathrm{sj}}^p,$$

(14)

where ${\xi }_k^p$ is the component of ${\mathit{\xi }}^p$ . Equations (11) and (12) following equation (14) can be written as:

$${\mathcal{W}}_{\mathrm{ext}}(P)={\int }_{\partial P}\mathbf{t}(\mathbf{n})·\stackrel{·}{\mathbf{u}}\mathrm{da}+{\int }_P\mathbf{b}(\mathbf{X},t)·\stackrel{·}{\mathbf{u}}\mathrm{dv}+{\int }_{\partial P}K(\mathbf{n}){\stackrel{·}{e}}^p\mathrm{da}$$

(15)

and

$${\mathcal{W}}_{\mathrm{int}}(P)={\int }_P[\mathbf{T}:{\stackrel{·}{\mathbf{H}}}^e+{\tau }^p{\stackrel{·}{e}}^p+{\mathit{\xi }}^p·\nabla {\stackrel{·}{e}}^p]\mathrm{dv}.$$

(16)

Let $\nu =(\tilde{\mathbf{u}},{\tilde{\mathbf{H}}}^p,{\tilde{e}}^p)$ be a list of virtual velocities consistent with:

$$\nabla \tilde{\mathbf{u}}={\tilde{\mathbf{H}}}^e+{\tilde{e}}^p{\mathbf{N}}^p.$$

(17)

The principle of virtual power states that:

$${\int }_{\partial P}\mathbf{t}(\mathbf{n})·\tilde{\mathbf{u}}\mathrm{da}+{\int }_P\mathbf{b}·\tilde{\mathbf{u}}\mathrm{dv}+{\int }_{\partial P}K(\mathbf{n}){\tilde{e}}^p\mathrm{da}={\int }_P[\mathbf{T}:{\tilde{\mathbf{H}}}^e+{\tau }^p{\tilde{e}}^p+{\mathit{\xi }}^p·\nabla {\tilde{e}}^p]\mathrm{dv}.$$

(18)

For a macroscopic motion, it is required that ${\tilde{e}}^p=0$ , so that ${\tilde{\mathbf{H}}}^e=\nabla \tilde{\mathbf{u}}$ . Equation (18) becomes:

$${\int }_{\partial P}\mathbf{t}(\mathbf{n})·\tilde{\mathbf{u}}\mathrm{da}-{\int }_{\partial P}\mathbf{Tn}·\tilde{\mathbf{u}}\mathrm{da}={\int }_P(\mathrm{div}\mathbf{T}+\mathbf{b})·\tilde{\mathbf{u}}\mathrm{dv}.$$

By the fundamental lemma on calculus of variation, we have:

• The macroscopic force balance:

$$\mathrm{div}\mathbf{T}+\mathbf{b}=0\mathrm{with}\mathbf{t}(\mathbf{n})=\mathbf{Tn}.$$

(19)

For a microscopic motion, it is required that $\tilde{\mathbf{u}}=0$ , which implies that ${\tilde{\mathbf{H}}}^e=-{\tilde{e}}^p{\mathbf{N}}^p$ . Equation (18) becomes:

$${\int }_{\partial P}(\mathbf{K}(\mathbf{n})-{\mathit{\xi }}^p·\mathbf{n}){\tilde{e}}^p\mathrm{da}={\int }_P({\tau }^p-\mathbf{T}:\mathbf{N}+\mathrm{div}{\mathit{\xi }}^p){\tilde{e}}^p\mathrm{dv}.$$

By the fundamental lemma on calculus of variation, we have:

• The microscopic force balance:

$$\tau ={\tau }^p-\mathrm{div}{\mathit{\xi }}^p\mathrm{with}K(\mathbf{n})={\mathit{\xi }}^p·\mathbf{n},$$

(20)

where $\tau =\mathbf{T}:{\mathbf{N}}^p$ .

4. Species transport and balance

Let $n^{\alpha }$ be species density measured mass per unit volume for different species $\alpha =1,2,\dots ,N$ . Let ${\mathbf{h}}^{\alpha }$ be species flux through the boundary $\partial P$ of the sub-body $P$ and with outward unit normal $\mathbf{n}$ . The flux ${\mathbf{h}}^{\alpha }$ is measured mass per unit time per unit area. Also, let $h^{\alpha }$ be species supply to the sub-body $P$ measured mass per unit time per unit volume. The global species balance states that:

$$\frac{d}{\mathrm{dt}}{\int }_P{n}^{\alpha }\mathrm{dv}=-{\int }_{\partial P}{\mathbf{h}}^{\alpha }·\mathbf{n}\mathrm{da}+{\int }_P{h}^{\alpha }\mathrm{dv}.$$

(21)

For small deformation in which the current and reference configurations are approximately the same, and applying the Gauss divergence theorem, the global species balance can be written as:

$${\int }_P({\stackrel{·}{n}}^{\alpha }+\mathrm{div}{\mathbf{h}}^{\alpha }-h^{\alpha })\mathrm{dv}=0.$$

Since $P$ is arbitrary, we have the species balance in local form as:

$${\stackrel{·}{n}}^{\alpha }=-\mathrm{div}{\mathbf{h}}^{\alpha }+h^{\alpha }.$$

(22)

5. Free-energy imbalance (second law of thermodynamics)

Increase in the entropy of a system is not less than the flow of entropy into the system. In the case of coupled species transport with plasticity under isothermal condition, the temporal increase in the entropy is non-negative. The free energy $\psi$ —measured per unit volume—of the system is related to the entropy and the internal energy through the Gibbs relation [7]. The second law of thermodynamics in terms of the free energy $\psi$ for a coupled species transport with plasticity states that the temporal increase in the free energy over $P$ is not greater than the sum of the external power expended on $P$ and energy flow due to species transport into $P$ . This is written mathematically as:

$${\int }_P\stackrel{·}{\psi }\mathrm{dv}\le {\mathcal{W}}_{\mathrm{ext}}(P)+\sum _{\alpha =1}^N[-{\int }_{\partial P}{\mu }^{\alpha }{\mathbf{h}}^{\alpha }·\mathbf{n}\mathrm{da}+{\int }_P{\mu }^{\alpha }{h}^{\alpha }\mathrm{dv}],$$

(23)

where ${\mu }^{\alpha }$ is the chemical potential for each $\alpha =1,2,\dots ,N$ . It is noted that:

$${\int }_{\partial P}{\mu }^{\alpha }{\mathbf{h}}^{\alpha }·\mathbf{n}\mathrm{da}={\int }_P{\mu }^{\alpha }\mathrm{div}{\mathbf{h}}^{\alpha }\mathrm{dv}+{\int }_P{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha }\mathrm{dv}.$$

By the power balance ${\mathcal{W}}_{\mathrm{ext}}(P)={\mathcal{W}}_{\mathrm{int}}(P)$ and the species balance equation (22), the inequality (23) becomes:

$${\int }_P[\stackrel{·}{\psi }-\mathbf{T}:{\stackrel{·}{\mathbf{E}}}^e-{\tau }^p{\stackrel{·}{e}}^p-{\mathit{\xi }}^p·\nabla {\stackrel{·}{e}}^p-\sum _{\alpha =1}^N({\mu }^{\alpha }{\stackrel{·}{n}}^{\alpha }-{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha })]\mathrm{dv}\le 0.$$

(24)

Since $P$ is arbitrary, the local free-energy imbalance from equation (24) is:

$$\stackrel{·}{\psi }-\mathbf{T}:{\stackrel{·}{\mathbf{E}}}^e-{\tau }^p{\stackrel{·}{e}}^p-{\mathit{\xi }}^p·\nabla {\stackrel{·}{e}}^p-\sum _{\alpha =1}^N({\mu }^{\alpha }{\stackrel{·}{n}}^{\alpha }-{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha })\le 0.$$

(25)

6. Constitutive relations

6.1. Constitutive relations for macroscopic stress, energetic microscopic stresses, and chemical potential

Suppose the defect energy is a function of the Burgers tensor and the accumulated plastic strain, then consistent with the work of Gurtin [25], it is shown by Borokinni and Ajayi [39] that the microscopic stress ${\mathbf{S}}^p$ which is power-conjugate to the Burgers tensor rate $\stackrel{·}{\mathbf{G}}$ is energetic. This implies that the vector microscopic stress $\mathit{\xi }$ expressed in equation (14) is purely energetic. Since the defect energy is also a function of $e^p$ , we shall assume that the scalar microscopic stress ${\tau }^p$ admits the additive decomposition:

$${\tau }^p={\tau }_{\mathrm{en}}^p+{\tau }_{\mathrm{dis}}^p,$$

(26)

into energetic part ${\tau }_{\mathrm{en}}^p$ and dissipative part ${\tau }_{\mathrm{dis}}^p$ . This additive decomposition accounts for backstress associated with the microscopic plastic stress. To capture dependence on the species densities, we shall assume that the free energy $\psi$ has the constitutive form and admits the separable form (into elastic free energy ${\psi }^e$ and defect energy ${\psi }^p$ ):

$$\begin{array}{c}\psi =\hat{\psi }({\mathbf{E}}^p,\mathbf{G},e^p,\overrightarrow{n})=\overline{\psi }({\mathbf{E}}^p,\nabla e^p,e^p,\overrightarrow{n}),\hfill \\ \psi ={\hat{\psi }}^e({\mathbf{E}}^e,\overrightarrow{n})+{\hat{\psi }}^p(\mathbf{G},e^p,\overrightarrow{n}),\hfill \end{array}$$

(27)

respectively, where $\overrightarrow{n}$ is the list $(n^1,n^2,\dots ,n^N)$ .

Furthermore, we shall assume the defect energy ${\psi }^p$ has the separable form:

$${\hat{\psi }}^p(\mathbf{G},e^p,\overrightarrow{n})={\hat{\psi }}^a(e^p,\overrightarrow{n})+{\hat{\psi }}^g(\nabla e^p,\overrightarrow{n}),$$

(28)

where ${\psi }^a$ and ${\psi }^g$ are the defect energies associated with accumulated plastic strain $e^p$ and gradient of the accumulated plastic strain $\nabla e^p$ , respectively. The constitutive relations for the Cauchy stress $\mathbf{T}$ , vector microscopic stress ${\mathit{\xi }}^p$ , energetic scalar microscopic stress ${\tau }_{\mathrm{en}}^p$ , and the chemical potential ${\mu }^{\alpha }$ are assumed as:

$$\begin{array}{c}\mathbf{T}=\hat{\mathbf{T}}({\mathbf{E}}^e,\overrightarrow{n}),{\mathit{\xi }}^p={\mathit{\xi }}^p(\nabla e^p,\overrightarrow{n}),\hfill \\ \multicolumn{1}{c}{{\tau }_{\mathrm{en}}^p={\hat{\tau }}_{\mathrm{en}}^p(e^p,\overrightarrow{n}){\mu }^{\alpha }={\hat{\mu }}^{\alpha }({\mathbf{E}}^e,e^p,\nabla e^p,\overrightarrow{n}){\mathbf{h}}^{\alpha }={\mathbf{h}}^{\alpha }({\mathbf{E}}^e,e^p,\nabla e^p,\overrightarrow{n},\nabla \overrightarrow{\mu }),}\end{array}$$

(29)

where $\nabla \overrightarrow{\mu }$ is the list $(\nabla {\mu }^1,\nabla {\mu }^2,\dots ,\nabla {\mu }^N)$ .

The free-energy imbalance equation (25) can be written as:

$$\begin{array}{c}(\frac{\partial {\hat{\psi }}^e({\mathbf{E}}^e,\overrightarrow{n})}{\partial {\mathbf{E}}^e}-\hat{\mathbf{T}}({\mathbf{E}}^e,\overrightarrow{n})):{\stackrel{·}{\mathbf{E}}}^e+(\frac{\partial {\hat{\psi }}^a(e^p,\overrightarrow{n})}{\partial e^p}-{\hat{\tau }}_{\mathrm{en}}^p(e^p,\overrightarrow{n})){\stackrel{·}{e}}^p+(\frac{\partial {\hat{\psi }}^g(\nabla e^p,\overrightarrow{n})}{\partial \nabla e^p}-{\hat{\mathit{\xi }}}^p(\nabla e^p,\overrightarrow{n}))·\nabla {\stackrel{·}{e}}^p\hfill \\ \multicolumn{1}{c}{+\sum _{\alpha =1}^N(\frac{\partial \hat{\psi }({\mathbf{E}}^e,\mathbf{G},\overrightarrow{n})}{\partial n^{\alpha }}-{\hat{\mu }}^{\alpha }({\mathbf{E}}^e,e^p,\nabla e^p,\overrightarrow{n})){\stackrel{·}{n}}^{\alpha }-{\tau }_{\mathrm{dis}}^p{\stackrel{·}{e}}^p+\sum _{\alpha =1}^N{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha }\le 0.}\end{array}$$

(30)

By the Coleman–Noll procedure, we have the following constitutive relations:

• Cauchy stress:

$$\mathbf{T}=\hat{\mathbf{T}}({\mathbf{E}}^e,\overrightarrow{n})=\frac{\partial {\hat{\psi }}^e({\mathbf{E}}^e,\overrightarrow{n})}{\partial {\mathbf{E}}^e},$$

(31)

• Vector microscopic stress:

$${\mathit{\xi }}^p={\hat{\mathit{\xi }}}^p(\nabla e^p,\overrightarrow{n})=\frac{\partial {\hat{\psi }}^g(\nabla e^p,\overrightarrow{n})}{\partial \nabla e^p},$$

(32)

• Scalar energetic microscopic stress:

$${\tau }_{\mathrm{en}}^p={\hat{\tau }}_{\mathrm{en}}^p(e^p,\overrightarrow{n})=\frac{\partial {\hat{\psi }}^a(e^p,\overrightarrow{n})}{\partial e^p},$$

(33)

• Chemical potential:

$${\mu }^{\alpha }={\hat{\mu }}^{\alpha }({\mathbf{E}}^e,e^p,\nabla e^p,\overrightarrow{n})=\frac{\partial \hat{\psi }({\mathbf{E}}^e,\mathbf{G},\overrightarrow{n})}{\partial n^{\alpha }},\alpha =1,2,\dots ,N.$$

(34)

Hence, the free-energy imbalance reduces to the dissipation inequality:

$${\tau }_{\mathrm{dis}}^p{\stackrel{·}{e}}^p-\sum _{\alpha =1}^N{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha }\ge 0,$$

(35)

for all possible ${\stackrel{·}{e}}^p$ and list $\{\nabla {\mu }^1,\nabla {\mu }^2,\dots ,\nabla {\mu }^N\}$ which are independent, and as a result, we have the species transport inequality:

$$\sum _{\alpha =1}^N{\mathbf{h}}^{\alpha }·\nabla {\mu }^{\alpha }\le 0,$$

(36)

and the mechanical dissipation inequality:

$${\tau }_{\mathrm{dis}}^p{\stackrel{·}{e}}^p\ge 0,$$

(37)

for all ${\stackrel{·}{e}}^p$ and list $\{\nabla {\mu }^1,\nabla {\mu }^2,\dots ,\nabla {\mu }^N\}$ .

Following the constitutive equations (31)–(34), the time derivatives for the stresses and chemical potential are given as follows:

$$\stackrel{·}{\mathbf{T}}=\frac{\partial \mathbf{T}}{\partial {\mathbf{E}}^e}{\stackrel{·}{\mathbf{E}}}^e+\sum _{\alpha =1}^N\frac{\partial \mathbf{T}}{\partial n^{\alpha }}{\stackrel{·}{n}}^{\alpha },$$

(38)

$${\stackrel{·}{\mathit{\xi }}}^p=\frac{\partial {\mathit{\xi }}^p}{\partial \nabla e^p}\nabla {\stackrel{·}{e}}^p+\sum _{\alpha =1}^N\frac{\partial {\mathit{\xi }}^p}{\partial n^{\alpha }}{\stackrel{·}{n}}^{\alpha },$$

(39)

$${\stackrel{·}{\tau }}_{\mathrm{en}}^p=\frac{\partial {\tau }_{\mathrm{en}}^p}{\partial e^p}{\stackrel{·}{e}}^p+\sum _{\alpha }^N\frac{\partial {\tau }_{\mathrm{en}}^p}{\partial n^{\alpha }}{\stackrel{·}{n}}^{\alpha },$$

(40)

and

$${\stackrel{·}{\mu }}^{\alpha }=\frac{\partial {\mu }^{\alpha }}{\partial {\mathbf{E}}^e}:{\stackrel{·}{\mathbf{E}}}^e+\frac{\partial {\tau }_{\mathrm{en}}^p}{\partial n^{\alpha }}{\stackrel{·}{e}}^p+\frac{\partial {\mathit{\xi }}^p}{\partial n^{\alpha }}·\nabla {\stackrel{·}{e}}^p+\sum _{\beta =1}^N\frac{\partial {\mu }^{\alpha }}{\partial n^{\beta }}{\stackrel{·}{n}}^{\beta }.$$

(41)

The elasticity tensor $\mathbb{C}$ , chemistry-elastic strain tensor ${\mathbf{S}}_e^{\alpha }$ ,and chemistry modulus ${\mathrm{\Lambda }}^{\alpha \beta }$ are defined by:

$$\mathbb{C}=\frac{\partial \hat{\mathbf{T}}({\mathbf{E}}^e,n^{\alpha })}{\partial {\mathbf{E}}^e},{\mathbf{S}}_e^{\alpha }=\frac{\partial {\hat{\mu }}^{\alpha }({\mathbf{E}}^e,\nabla e^p,e^p,\overrightarrow{n})}{\partial {\mathbf{E}}^e},\mathrm{and}{\mathrm{\Lambda }}^{\alpha \beta }=\frac{\partial {\hat{\mu }}^{\alpha }({\mathbf{E}}^e,\nabla e^p,e^p,\overrightarrow{n})}{\partial n^{\beta }}.$$

(42)

We introduce scalar chemistry-plastic strain ${\varphi }^{\alpha }$ and vector chemistry-plastic strain gradient ${\mathit{\zeta }}^{\alpha }$ defined as:

$${\varphi }^{\alpha }=\frac{\partial {\hat{\tau }}_{\mathrm{en}}^p(e^p,\overrightarrow{n})}{\partial n^{\alpha }}\mathrm{and}{\mathbf{\zeta }}^{\alpha }=\frac{\partial {\hat{\mathit{\xi }}}^p(\nabla e^p,\overrightarrow{n})}{\partial n^{\alpha }}.$$

(43)

The scalar $\omega$ and tensor $\mathbf{B}$ associated with plastic strain and plastic strain gradient are defined as:

$$\omega =\frac{\partial {\tau }_{\mathrm{en}}^p}{\partial e^p}\mathrm{and}\mathbf{B}=\frac{\partial {\mathit{\xi }}^p}{\partial \nabla e^p},$$

(44)

respectively.

In small deformation theory, the reference configuration is assumed to be natural with respect to some fixed species density list ${\overrightarrow{n}}_o$ . In this configuration, the free energy is a minimum at ${\mathbf{E}}^e=0$ . In addition, we assume that for a reference configuration that is natural with respect to ${\overrightarrow{n}}_0$ , the free energy is minimum at ${\mathbf{E}}^e=0$ , $e^p=0$ , and $\nabla e^p=0$ . Given a field $\mathrm{\Phi }=\mathrm{\Phi }({\mathbf{E}}^e,\nabla e^p,e^p,\overrightarrow{n})$ , we define $\mathrm{\Phi }{|}_0$ by:

$$\mathrm{\Phi }{|}_0:=\mathrm{\Phi }(0,0,0,{\overrightarrow{n}}_0).$$

(45)

It can be assumed that $\psi {|}_0=0$ . Following equations (27) and (28), the free energy can be expanded as:

$$\begin{array}{c}\overline{\psi }({\mathbf{E}}^e,\nabla e^p,e^p,\overrightarrow{n})=\mathbf{T}{|}_0:{\mathbf{E}}^e+{\tau }_{\mathrm{en}}^p{|}_0{e}^p+{\mathit{\xi }}^p{|}_0·\nabla e^p+\sum _{\alpha =1}^N{\mu }^{\alpha }{|}_0(n^{\alpha }-n_0^{\alpha })+\hfill \\ \multicolumn{1}{c}{\frac{1}{2}[(\mathbb{C}{|}_0{\mathbf{E}}^e):{\mathbf{E}}^e+\omega {|}_0{e}^p{e}^p+(\mathbf{B}{|}_0\nabla e^p)·\nabla e^p+\sum _{\alpha ,\beta =1}^N{\mathrm{\Lambda }}^{\alpha \beta }{|}_0(n^{\alpha }-n_0^{\alpha })(n^{\beta }-n_0^{\beta })]+}\\ \multicolumn{1}{c}{{\mathbf{E}}^e:\sum _{\alpha }^N{\mathbf{S}}_e^{\alpha }{|}_0(n^{\alpha }-n_0^{\alpha })+\sum _{\alpha =1}^N{\varphi }^{\alpha }{|}_0(n^{\alpha }-n_0^{\alpha })e^p+\nabla e^p·\sum _{\alpha =1}^N{\mathbf{\zeta }}^{\alpha }{|}_0(n^{\alpha }-n_0^{\alpha })+o({ϵ}^2),}\end{array}$$

(46)

as $ϵ\to 0$ , where for a length scale $q$ :

$$ϵ=\sqrt{|{\mathbf{E}}^e{|}^2+|e^p{|}^2+q^2|\nabla e^p{|}^2+\sum _{\alpha =1}^N{\left|\frac{n^{\alpha }-n_0^{\alpha }}{n_0^{\alpha }}\right|}^2},$$

(47)

is assumed small.

Since the free energy is minimum in a natural reference configuration at $(0,0,0,{\overrightarrow{n}}_0)$ , then we have:

$$\mathbf{T}{|}_0=0,{\tau }_{\mathrm{en}}^p{|}_0=0,{\mathit{\xi }}^p{|}_0=0,{\mu }^{\alpha }{|}_0=0.$$

(48)

Neglecting higher-order term in the expansion of $\mathbb{C}$ , $\omega$ , $\mathbf{B}$ , ${\mathrm{\Lambda }}^{\alpha \beta }$ , ${\mathbf{S}}_e^{\alpha }$ , ${\varphi }^{\alpha }$ , and ${\mathbf{\zeta }}^{\alpha }$ in the sense:

$$\mathbb{C}{|}_0=\mathbb{C},\omega {|}_0=\omega ,\mathbf{B}{|}_0=\mathbf{B},{\mathrm{\Lambda }}^{\alpha \beta }{|}_0={\mathrm{\Lambda }}^{\alpha \beta },{\mathbf{S}}_e^{\alpha }{|}_0={\mathbf{S}}_e^{\alpha },{\varphi }^{\alpha }{|}_0={\varphi }^{\alpha },\mathrm{and}{\mathbf{\zeta }}^{\alpha }{|}_0={\mathbf{\zeta }}^{\alpha },$$

(49)

then neglecting $o({ϵ}^2)$ , the free energy becomes:

$$\begin{array}{c}\overline{\psi }({\mathbf{E}}^e,\nabla e^p,e^p,\overrightarrow{n})=\frac{1}{2}{\mathbf{E}}^e:\mathbb{C}{\mathbf{E}}^e+\frac{1}{2}\omega |e^p{|}^2+\nabla e^p·\mathbf{B}\nabla e^p+\frac{1}{2}\sum _{\alpha ,\beta =1}^N{\mathrm{\Lambda }}^{\alpha \beta }(n^{\alpha }-n_0^{\alpha })(n^{\beta }-n_0^{\beta })+\hfill \\ \multicolumn{1}{c}{\sum _{\alpha =1}^N({\mathbf{S}}_e^{\alpha }:{\mathbf{E}}^e)(n^{\alpha }-n_0^{\alpha })+\sum _{\alpha =1}^N{\varphi }^{\alpha }{e}^p(n^{\alpha }-n_0^{\alpha })+\sum _{\alpha =1}^N({\mathbf{\zeta }}^{\alpha }·\nabla e^p)(n^{\alpha }-n_0^{\alpha }).}\end{array}$$

(50)

Using equations (31)–(34) and (50), we have the following constitutive relations:

• Cauchy stress:

$$\hat{\mathbf{T}}({\mathbf{E}}^e,\overrightarrow{n})=\mathbb{C}{\mathbf{E}}^e+\sum _{\alpha }^N(n^{\alpha }-n_0^{\alpha }){\mathbf{S}}_e^{\alpha };$$

(51)

• Energetic microscopic stresses:

$${\hat{\mathit{\xi }}}^p(\nabla e^p,\overrightarrow{n})=\mathbf{B}\nabla e^p+\sum _{\alpha =1}^N(n^{\alpha }-n_0^{\alpha }){\mathbf{\zeta }}^{\alpha };$$

(52)

$${\hat{\tau }}_{\mathrm{en}}^p(e^p,\overrightarrow{n})=\omega e^p+\sum _{\alpha =1}^N(n^{\alpha }-n_0^{\alpha }){\varphi }^{\alpha };$$

(53)

• Chemical potential for each species $\alpha$ :

$${\hat{\mu }}^{\alpha }({\mathbf{E}}^e,e^p,\nabla e^p,\overrightarrow{n})=\sum _{\beta =1}^N{\mathrm{\Lambda }}^{\alpha \beta }(n^{\alpha }-n_0^{\alpha })+{\mathbf{S}}_e^{\alpha }:{\mathbf{E}}^e+{\varphi }^{\alpha }{e}^p+{\mathbf{\zeta }}^{\alpha }·\nabla e^p.$$

(54)

6.2. Constitutive relation for species flux

We assume the species flux ${\mathbf{h}}^{\alpha }$ obeys Fick’s law:

$${\mathbf{h}}^{\alpha }=-\sum _{\beta =1}^N{\mathbf{M}}^{\alpha \beta }\nabla {\mu }^{\beta },$$

(55)

where ${\mathbf{M}}^{\alpha \beta }$ is the mobility tensor. Consistent with the small deformation theory, we assume that ${\mathbf{M}}^{\alpha \beta }$ is a constant tensor. It is worthy of note that since the chemical potential has the constitutive form equation (54), then the chemical potential gradient $\nabla {\mu }^{\beta }$ can be written in component form as:

$${\mu }_{,i}^{\beta }=\frac{\partial {\mu }^{\beta }}{\partial E_{\mathrm{jk}}^e}{E}_{\mathrm{jk},i}^e+\frac{\partial {\mu }^{\beta }}{\partial e^p}{e}_{,i}^p+\frac{\partial {\mu }^{\beta }}{\partial e_{,k}^p}{e}_{\mathrm{ki}}^p+\sum _{\gamma =1}^N\frac{\partial {\mu }^{\beta }}{\partial n^{\gamma }}{n}_{,i}^{\gamma }.$$

By letting the invariant form of the component $\frac{\partial {\mu }^{\beta }}{\partial E_{\mathrm{jk}}^e}{E}_{\mathrm{jk},i}^e$ be the vector ${\mathbf{S}}_e^{\beta }:\nabla {\mathbf{E}}^e$ , then the gradient of the chemical potential $\nabla {\mu }^{\beta }$ can be written as:

$$\nabla {\mu }^{\beta }={\mathbf{S}}_e^{\beta }:\nabla {\mathbf{E}}^e+{\varphi }^{\beta }\nabla e^p+(\nabla \nabla e^p){\mathbf{\zeta }}^{\beta }+\sum _{\gamma =1}^N{\mathrm{\Lambda }}^{\beta \gamma }\nabla n^{\gamma }.$$

(56)

Hence, the constitutive relation for the species flux is given as:

$${\mathbf{h}}^{\alpha }=-\sum _{\beta =1}^N{\mathbf{M}}^{\alpha \beta }({\mathbf{S}}_e^{\beta }:\nabla {\mathbf{E}}^e+{\varphi }^{\beta }\nabla e^p+(\nabla \nabla e^p){\mathbf{\zeta }}^{\beta }+\sum _{\gamma =1}^N{\mathrm{\Lambda }}^{\beta \gamma }\nabla n^{\gamma }).$$

(57)

6.3. Dissipative microscopic stress. Mises–Hill flow equations

We take as constitutive assumption the relation:

$${\tau }_{\mathrm{dis}}^p={\hat{\tau }}_{\mathrm{dis}}^p(e^p,{\mathbf{N}}^p,\overrightarrow{n}),$$

(58)

for the dissipative microscopic stress. Here, the plastic flow is assumed to be rate-independent so that the accumulated plastic strain $e^p$ is also a hardening variable satisfying equation (5).

The mechanical dissipation $\delta$ for this study is defined as (see equation (37)):

$$\delta =\hat{\delta }(e^p,{\stackrel{·}{e}}^p,{\mathbf{N}}^p,\overrightarrow{n})={\hat{\tau }}_{\mathrm{dis}}^p(e^p,{\mathbf{N}}^p,\overrightarrow{n}){\stackrel{·}{e}}^p.$$

(59)

Consistent with the second law of thermodynamics, it is noted that $\delta >0$ for ${\stackrel{·}{e}}^p\ne 0$ and ${\tau }_{\mathrm{dis}}^p\ne 0.$ The flow resistance $Y=\hat{Y}(e^p,{\mathbf{N}}^p,\overrightarrow{n})$ is defined as the mechanical plastic dissipation per unit accumulated plastic strain rate, i.e.,

$$Y=\hat{Y}(e^p,{\mathbf{N}}^p,\overrightarrow{n})=\frac{\hat{\delta }(e^p,{\mathbf{N}}^p,\overrightarrow{n})}{{\stackrel{·}{e}}^p}\mathrm{for}\mathrm{all}{\stackrel{·}{e}}^p\ne 0.$$

(60)

Equations (58)–(60) imply that the dissipative scalar microscopic stress satisfies the flow equation:

$${\hat{\tau }}_{\mathrm{dis}}^p(e^p,{\mathbf{N}}^p,\overrightarrow{n})=\hat{Y}(e^p,{\mathbf{N}}^p,\overrightarrow{n})\mathrm{if}{\stackrel{·}{e}}^p\ne 0.$$

Assume strong isotropy, then the flow resistance would be independent of the flow direction. Consequences of this assumption are:

$${\tau }_{\mathrm{dis}}^p={\hat{\tau }}_{\mathrm{dis}}^p(e^p,\overrightarrow{n})=Y(e^p,\overrightarrow{n})\mathrm{if}{\stackrel{·}{e}}^p\ne 0\mathrm{and}\delta =\hat{\delta }(e^p,{\stackrel{·}{e}}^p,\overrightarrow{n})=Y(e^p,\overrightarrow{n}){\stackrel{·}{e}}^p.$$

(61)

Equation (61) is the Mises–Hill flow equation for a rate-independent plastic flow.

The elastic range and the no-flow condition are given by:

$$|{\tau }_{\mathrm{dis}}^p|\le \hat{Y}(e^p,\overrightarrow{n})\mathrm{for}\mathrm{all}{\stackrel{·}{e}}^p$$

(62)

and

$${\stackrel{·}{e}}^p=0\mathrm{if}|{\tau }_{\mathrm{dis}}^p|<\hat{Y}(e^p,\overrightarrow{n}),$$

(63)

respectively.

7. Governing equations for gradient plasticity with species transport

Following the constitutive relation equation (51), the macroscopic force balance equation (19) can be written in the form:

$$\mathrm{div}[\mathbb{C}(\mathbf{E}-{\mathbf{E}}^p)+\sum _{\alpha =1}^N(n^{\alpha }-n_0^{\alpha }){\mathbf{S}}_e^{\alpha }]+\mathbf{b}=0,$$

(64)

where:

$$\mathbf{E}=\frac{1}{2}[\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\mathrm{T}}].$$

The microscopic force balance equation (20) using the constitutive relation equations (52) and (53) and plastic flow law equation (61) can be written as:

$$\tau +\mathbf{B}:\nabla \nabla e^p-\omega e^p+\sum _{\alpha =1}^N({\mathbf{\zeta }}^{\alpha }·\nabla n^{\alpha }-{\varphi }^{\alpha }(n^{\alpha }-n_0^{\alpha }))=\hat{Y}(e^p,\overrightarrow{n}).$$

(65)

Define chemical, elastic, plastic, and plastic-gradient diffusivities as:

$${\mathbf{D}}_e^{\alpha \gamma }=\sum _{\beta =1}^N{\mathbf{M}}^{\alpha \beta }{\mathrm{\Lambda }}^{\beta \gamma },{\mathbf{J}}^{\alpha }=\sum _{\beta =1}^N{\mathbf{M}}^{\alpha \beta }{\mathbf{S}}_e^{\beta },{\mathbf{L}}_p^{\alpha }=\sum _{\beta }^N{\mathbf{M}}^{\alpha \beta }{\varphi }^{\beta },\mathrm{and}{\mathbf{g}}_p^{\alpha }=\sum _{\beta =1}^N{\mathbf{M}}^{\alpha \beta }{\mathbf{\zeta }}^{\beta },$$

(66)

respectively. Using equations (57) and (66), the divergence of the species flux takes the form:

$$\mathrm{div}{\mathbf{h}}^{\alpha }=-(\sum _{\beta =1}^N{\mathbf{D}}^{\alpha \beta }:\nabla \nabla n^{\beta })-{\mathbf{L}}_p^{\alpha }:\nabla \nabla e^p-{\mathbf{g}}_p^{\alpha }·\nabla \mathrm{\Delta }{e}^p-{\mathbf{J}}^{\alpha }:\mathrm{\Delta }(\mathbf{E}-{\mathbf{E}}^p).$$

(67)

Hence, the species balance equation (22) can be written as:

$${\stackrel{·}{n}}^{\alpha }=(\sum _{\beta =1}^N{\mathbf{D}}^{\alpha \beta }:\nabla \nabla n^{\beta })+{\mathbf{L}}_p^{\alpha }:\nabla \nabla e^p+{\mathbf{g}}_p^{\alpha }·\nabla \mathrm{\Delta }{e}^p+{\mathbf{J}}^{\alpha }:\mathrm{\Delta }(\mathbf{E}-{\mathbf{E}}^p)+h^{\alpha }.$$

(68)

8. Initial and boundary conditions of the governing equations

8.1. Initial conditions

We assume the following specification for the species density $n^{\alpha }$ for each $\alpha =1,2,\dots ,N$ and accumulated plastic strain at time $t=0$ :

$$n^{\alpha }(\mathbf{X},0)=n_{*}^{\alpha }(\mathbf{X})\mathrm{and}{e}^p(\mathbf{X},0)=0\mathrm{for}\mathrm{all}\mathbf{X}\in B.$$

(69)

where $n_{*}^{\alpha }$ is given for each $\alpha$ .

8.2. Boundary condition for the macroscopic force balance

Let $\partial B=\partial B_u\cup \partial B_t$ be the union of two complementary sub-surfaces $\partial B_u$ and $\partial B_t$ such that the displacement $\mathbf{u}$ and the traction $\mathbf{t}(\mathbf{n})=\mathbf{Tn}$ are specified on $\partial B_u$ and $\partial B_t$ , respectively, as follows:

$$\mathbf{u}={\mathbf{u}}_0\mathrm{on}\partial B_u\mathrm{and}\mathbf{Tn}={\mathbf{t}}_0\mathrm{on}\partial B_t.$$

(70)

8.3. Boundary condition for microscopic force balance

Based on plastic flow, it can be assumed that the body is the union of time-dependent complementary sub-bodies $B_e(t)$ and $B_p(t)$ known as elastic and plastic regions, respectively, intersecting at an elastic–plastic interface $I(t)$ . The external plastic boundary $S(t)$ is defined as:

$$S(t)=\partial B_p(t)\cap \partial B,$$

(71)

where $\partial B_p(t)$ is the boundary of the region $B_p(t)$ . Also, the boundary $\partial B$ can be written as the union of the complementary sub-surfaces $\partial B_{\mathrm{hard}}$ and $\partial B_{\mathrm{free}}$ characterizing microscopically hard and microscopically free portions of $\partial B$ interacting with the environment. We shall limit this study to conditions in which $\partial B_{\mathrm{hard}}$ is a barrier to flow of dislocations and $\partial B_{\mathrm{free}}$ allows flow of dislocations. The interactions of the boundaries $\partial B_{\mathrm{hard}}$ and $\partial B_{\mathrm{free}}$ with $S(t)$ are expressed through the complementary sub-surfaces:

$$S_{\mathrm{hard}}(t)=\partial B_{\mathrm{hard}}\cap S(t)S_{\mathrm{free}}(t)=\partial B_{\mathrm{free}}\cap S(t),$$

(72)

of $S(t)$ . We consider the following boundary conditions:

$$e^p(\mathbf{X},t)=0\mathrm{on}{S}_{\mathrm{hard}}(t)\mathrm{and}{\mathit{\xi }}^p(\mathbf{X},t)·\mathbf{n}(\mathbf{X},t)=0\mathrm{on}{S}_{\mathrm{free}}(t).$$

(73)

We note here that the choices in equations (69)₂ and (73)₁ are consistent with the no flow condition:

$${\stackrel{·}{e}}^p(\mathbf{X},t)=0\mathrm{on}{S}_{\mathrm{hard}}(t).$$

(74)

8.4. Boundary condition for species balance

Assume that $\partial B$ is the union of the complementary sub-surfaces $\partial B_n^{\alpha }$ and $\partial B_h^{\alpha }$ for each $\alpha$ such that:

$$n^{\alpha }(\mathbf{X},t)=n_{**}^{\alpha }\mathrm{on}\partial B_n^{\alpha }\mathrm{and}{\mathbf{h}}^{\alpha }·\mathbf{n}=H^{\alpha }\mathrm{on}\partial B_h^{\alpha }\mathrm{for}\mathrm{each}\alpha .$$

(75)

9. Variational formulation of problem

Here, we follow the variational formulations used by Gurtin and Reddy [40] and Reddy [41]. We keep in mind that the mechanical dissipation $\delta$ is a function of $e^p$ , ${\stackrel{·}{e}}^p$ , and $\overrightarrow{n}$ ; for brevity, we write $\delta =\hat{\delta }({\stackrel{·}{e}}^p)$ . It is clear from equations (61) and (62) that:

$$\delta ({\tilde{e}}^p)\ge {\tau }_{\mathrm{dis}}^p{\tilde{e}}^p\mathrm{for}\mathrm{all}\mathrm{kinematically}\mathrm{admissible}{\tilde{e}}^p,$$

(76)

satisfying the initial and boundary conditions equations (69) and (73). A combination of equations (61) and (76) is equivalent to the inequality:

$$\delta ({\tilde{e}}^p)\ge \delta ({\stackrel{·}{e}}^p)+{\tau }_{\mathrm{dis}}^p({\tilde{e}}^p-{\stackrel{·}{e}}^p),$$

(77)

where ${\tau }_{\mathrm{dis}}^p$ satisfies:

$${\tau }_{\mathrm{dis}}^p={\hat{\tau }}_{\mathrm{dis}}^p(e^p,{\stackrel{·}{e}}^p)=\frac{\partial \hat{\delta }({\stackrel{·}{e}}^p)}{\partial {\stackrel{·}{e}}^p}\mathrm{for}{\stackrel{·}{e}}^p\ne 0.$$

(78)

Integrating equation (77) over the body $B$ , we have:

$${\int }_B[\delta ({\tilde{e}}^p-\delta ({\stackrel{·}{e}}^p))-{\tau }_{\mathrm{dis}}^p({\tilde{e}}^p-{\stackrel{·}{e}}^p)]\mathrm{dv}\ge 0.$$

(79)

The microscopic force balance equation (20) together with the boundary conditions equation (73) is equivalent to:

$${\int }_B[({\tau }_{\mathrm{en}}^p+{\tau }_{\mathrm{dis}}^p)({\tilde{e}}^p-{\stackrel{·}{e}}^p)+{\mathit{\xi }}^p·\nabla (\tilde{e}-{\stackrel{·}{e}}^p)-\tau (\tilde{e}-{\stackrel{·}{e}}^p)]\mathrm{dv}=0.$$

(80)

Adding equations (79) and (80), we obtain:

$${\int }_B[\delta ({\tilde{e}}^p)-\delta ({\stackrel{·}{e}}^p)+{\tau }_{\mathrm{en}}^p(\tilde{e}-{\stackrel{·}{e}}^p)+{\mathit{\xi }}^p·\nabla (\tilde{e}-{\stackrel{·}{e}}^p)-\tau (\tilde{e}-{\stackrel{·}{e}}^p)]\mathrm{dv}\ge 0.$$

(81)

The inequality equation (81) is the global variational inequality for microscopic force balance.

Let $\tilde{\mathbf{u}}$ be a kinematically admissible velocity in the sense:

$$\tilde{\mathbf{u}}={\stackrel{·}{\mathbf{u}}}_0\mathrm{on}\partial B_u.$$

The macroscopic force balance equation (19) together with the boundary conditions equation (72) is equivalent to:

$${\int }_B\mathbf{T}:(\tilde{\mathbf{E}}-\stackrel{·}{\mathbf{E}})\mathrm{dv}-{\int }_{\partial B_t}{\mathbf{t}}_0·(\tilde{\mathbf{u}}-\stackrel{·}{\mathbf{u}})\mathrm{da}-{\int }_B\mathbf{b}·(\tilde{\mathbf{u}}-\stackrel{·}{\mathbf{u}})\mathrm{dv}=0,$$

(82)

where $\mathbf{E}=\frac{1}{2}[\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\mathrm{T}}]$ .

Combining equations (81) and (82), we have the global variational inequality for macroscopic and microscopic force balances as:

$$\begin{array}{c}{\int }_B\delta ({\tilde{e}}^p)\mathrm{dv}-{\int }_B\delta ({\stackrel{·}{e}}^p)\mathrm{dv}+{\int }_B({\tau }_{\mathrm{en}}^p({\tilde{e}}^p-{\stackrel{·}{e}}^p)+{\mathit{\xi }}^p·\nabla ({\tilde{e}}^p-{\stackrel{·}{e}}^p))\mathrm{dv}+\hfill \\ {\int }_B\mathbf{T}:[(\tilde{\mathbf{E}}-{\tilde{\mathbf{E}}}^p)-(\stackrel{·}{\mathbf{E}}-{\stackrel{·}{\mathbf{E}}}^p)]\mathrm{dv}-{\int }_{\partial B_t}{\mathbf{t}}_0·(\tilde{\mathbf{u}}-\stackrel{·}{\mathbf{u}})\mathrm{da}-{\int }_B\mathbf{b}·(\tilde{\mathbf{u}}-\stackrel{·}{\mathbf{u}})\mathrm{dv}\ge 0,\hfill \end{array}$$

(83)

where ${\tilde{\mathbf{E}}}^p={\tilde{e}}^p{\mathbf{N}}^p$ .

Let ${\tilde{n}}^{\alpha }$ be an admissible species density in the sense that ${\tilde{n}}^{\alpha }=0$ on $\partial B_n^{\alpha }$ for each $\alpha$ . The species balance equation (22) together with boundary condition equation (75) is equivalent to the global variational equation:

$${\int }_B{\stackrel{·}{n}}^{\alpha }{\tilde{n}}^{\alpha }\mathrm{dv}+{\int }_{\partial B_h^{\alpha }}{\tilde{n}}^{\alpha }{H}^{\alpha }\mathrm{da}-{\int }_B{\mathbf{h}}^{\alpha }·\nabla {\tilde{n}}^{\alpha }\mathrm{dv}-{\int }_B{h}^{\alpha }{\tilde{n}}^{\alpha }\mathrm{dv}=0.$$

(84)

10. Isotropic strain-gradient plasticity with species transport

For isotropic homogeneous materials, there exist scalar constants $\mu$ , $\lambda$ , $m^{\alpha \beta }$ , $s^{\alpha }$ , and $k_1$ such that:

$$\mathbb{C}{\mathbf{E}}^e=2\mu {\mathbf{E}}^e+\lambda (\mathrm{tr}{\mathbf{E}}^e)\mathbf{I},{\mathbf{M}}^{\alpha \beta }=m^{\alpha \beta }\mathbf{I},{\mathbf{S}}^{\alpha }=s^{\alpha }\mathbf{I}\mathrm{and}\mathbf{B}=k_1\mathbf{I},$$

(85)

where $\mathbf{I}$ is the second-order identity tensor. The constants $\mu$ and $\lambda$ are the Lamés constants. $m^{\alpha \beta }$ and $s^{\alpha }$ are the mobility modulus and chemistry elastic modulus, respectively. $k_1$ is a positive constant.

Using equation (85):

• The macroscopic force balance equation (64) can be written as:

$$2\mu \mathrm{div}{\mathbf{E}}^e+\lambda \nabla (\mathrm{tr}{\mathbf{E}}^e)+\sum _{\alpha =1}^N{s}^{\alpha }\nabla n^{\alpha }+\mathbf{b}=0,$$

(86)

• The microscopic force balance equation (65) can be written as:

$$\tau +k_1\mathrm{\Delta }{e}^p-\omega e^p+\sum _{\alpha =1}^N({\mathbf{\zeta }}^{\alpha }·\nabla n^{\alpha }-{\varphi }^{\alpha }(n^{\alpha }-n_0^{\alpha }))=\hat{Y}(e^p,\overrightarrow{n}),$$

(87)

• The species balance equation (67) becomes:

$${\stackrel{·}{n}}^{\alpha }=\sum _{\beta =1}^N{m}^{\alpha \beta }[\left(\sum _{\gamma =1}^N{\mathrm{\Lambda }}^{\beta \gamma }\mathrm{\Delta }{n}^{\gamma }\right)+{\varphi }^{\beta }\mathrm{\Delta }{e}^p+{\mathbf{\zeta }}^{\alpha }·\nabla \mathrm{\Delta }{e}^p+s^{\beta }\mathrm{\Delta }\mathrm{tr}{\mathbf{E}}^e].$$

(88)

Equations (86)–(88) together with the initial-boundary conditions equations (69), (70), and (73)–(75) are to be solved for the displacement $\mathbf{u}$ , the accumulated plastic strain $e^p$ , and the species density $n^{\alpha }$ for each $\alpha$ and given a flow direction ${\mathbf{N}}^p$ for an isotropic material.

11. Conclusion

Within the context of small deformations, this work presents a continuum mechanical coupled theory of strain-gradient plasticity with multiple species transport under thermodynamic restrictions for isotropic solids. The scalar chemistry-plastic strain and vector chemistry-plastic strain gradients are natural deductions arising from the interactions of species with gradient plastic materials.