1 Introduction
As nowadays experimental techniques reveal properties of materials (e.g., of cells [
1]) on the submicron scale, a robust theory is needed to describe such phenomena. While atomistic simulations still cannot describe the observations on such scales over a sufficiently extended period of time, continuum approaches turn out to be too coarse, due to their inability of describing the fluctuations observed in experiments, e.g., on bio-membranes [
2]. The current manuscript provides a theory suitable for such scales, by providing an atomistic definition of the elastic distortion, as well as its reversible, irreversible, and stochastic dynamics.
Inelastic behavior is intrinsic to practically all materials. While the reasons for such behavior differ significantly between materials, there are also common aspects, particularly when it comes to the general structure of the modeling approaches. In principle, the current stress state depends on the entire history of imposed deformation (e.g., by way of a memory function/kernel), which is at the basis of Boltzmann’s principle of superposition [
3,
4] used to describe linear viscoelasticity. However, if nonlinearity comes into play, a different approach is often taken. A prominent alternative is to decompose the total deformation into its elastic and inelastic contributions. In the field of solids, this is realized by the multiplicative decomposition of the total deformation gradient
\(\textbf{F}\), according to Kröner [
5] and Lee and Liu [
6,
7], which is a cornerstone for modeling finite deformation plasticity (see, e.g., [
8‐
13]): It uses
\(\textbf{F}= \textbf{F}_{\textrm{e}} \cdot \textbf{F}_{\textrm{p}}\), where
\(\textbf{F}_{\textrm{e}}\) and
\(\textbf{F}_{\textrm{p}}\) represent the elastic and plastic parts of the total deformation gradient, respectively. The quantity
\(\textbf{F}_{\textrm{p}}\) maps line elements from the reference configuration to an intermediate stress-free configuration, from which
\(\textbf{F}_{\textrm{e}}\) maps these line elements to the current configuration. Conceptually,
\(\textbf{F}_{\textrm{e}}\) can be determined by the instantaneous (to avoid relaxation effects) unloading of a deformed sample. The main goal of such approaches is the formulation of an evolution equation for
\(\textbf{F}_{\textrm{e}}\) that contains both an affine contribution (representative of the kinematics of
\(\textbf{F}\)) and a contribution that describes the
rate of plastic deformation (representative of the inelastic dynamics of
\(\textbf{F}_{\textrm{p}}\)); notably, the plastic deformation
\(\textbf{F}_{\textrm{p}}\) itself is not involved, as long as, for example, strain/work hardening is not considered. A discussion of some critical aspects of the multiplicative decomposition, particularly in relation to the intermediate configuration, can be found in [
14]. A concept related to that of the intermediate configuration is the so-called natural reference state [
15]; for a discussion on similarities and differences between the two, the reader is referred to [
15]. In either case, the elastic deformation
\(\textbf{F}_{\textrm{e}}\) is the fundamental dynamic variable for modeling the material behavior under finite deformation.
Instead of the elastic deformation
\(\textbf{F}_{\textrm{e}}\), one could equivalently use the so-called distortion
\(\textbf{A}\), which is the inverse of
\(\textbf{F}_{\textrm{e}}\),
\(\textbf{A}= \textbf{F}_{\textrm{e}}^{-1}\). This is the approach taken in [
16‐
18]. Although the distortion and deformation gradient are equivalent in the purely elastic case, where
\(\textbf{A}= \textbf{F}^{-1}\), distortion becomes advantageous in the case of dissipation, where the distortion only reflects the elastic part of the deformation gradient,
\(\textbf{A}= \textbf{F}_{\textrm{e}}^{-1}\). Dissipation can actually break the continuum compatibility conditions: the configuration from which the elastic part of the deformation gradient is mapping, which is called the natural configuration in the distortion-related literature, ceases to exist as a global configuration [
19‐
22]. This happens, for instance, in the case of a screw dislocation [
23], where the natural configuration has a discontinuity, and thus, we cannot take derivatives with respect to it, and
\(\textbf{F}_{\textrm{e}}\) does not exist globally. In the case of dislocations, the multiplicative splitting can be substantiated mathematically by projection of the deformation tensors to a regular part and a part that contains the Dirac distribution [
14]. In contrast to the natural configuration, the current configuration always exists, so we can take derivatives with respect to it, and
\(\textbf{A}\) exists. If the distortion is curl-free, its contour integrals give the natural configuration, that represents the equilibrium state upon instantaneous unloading [
19,
20]. If, however, the distortion is not curl-free (for instance in case of dislocations, where the curl of the distortion equals the dislocation density tensor [
21,
24‐
27]), then the result of the integration depends on the particular path chosen, which means that the natural configuration is defined only locally.
Corresponding to the dynamics of the elastic deformation (e.g., see [
10,
11]), the dynamics of the distortion can be written in the general form [
17]
$$\begin{aligned} D_t \textbf{A}= & {} -\textbf{A}\cdot (\partial \textbf{v}/\partial \textbf{r}) + {}^{\textrm{p}\!}\textbf{L}\cdot \textbf{A} \end{aligned}$$
(1a)
$$\begin{aligned} {}= & {} -\textbf{A}\cdot (\partial \textbf{v}/\partial \textbf{r}) + \textbf{A}\cdot {}^{\textrm{p}\!}{\check{\textbf{L}}}, \end{aligned}$$
(1b)
where
\(D_t\) denotes the material derivative,
\(\textbf{v}\) is the velocity field, and
\({}^{\textrm{p}\!}\textbf{L}\) and
\({}^{\textrm{p}\!}{\check{\textbf{L}}}\) stand for the plastic deformation-rate tensors in the natural and current configurations, respectively.
If one disregards inelastic deformation, it is obvious that by virtue of its definition, the deformation gradient is in principle a Lagrangian field, with the
reference position as dummy field variable [
28,
29]. In contrast, the distortion is quite naturally an Eulerian field, with the
current position as dummy field variable. This being said, it is of course possible to, once defined, re-interpret the deformation gradient as well as its elastic and plastic parts as Eulerian fields (e.g., see [
30,
31]). Nevertheless, when aiming at an Eulerian formulation for modeling finite deformation inelastic behavior, the Eulerian distortion simplifies the description because it involves only one mapping (from the current to the reference configuration); in contrast, for instance, the Eulerian deformation gradient is the composition of two mappings (mapping from the reference to the current configuration, but position evaluation in the current configuration).
In this paper, the goal is to express the continuum Eulerian field of distortion in terms of the arrangement of the constituent microscopic particles of the material and to derive the distortion dynamics Eq. (
1) in relation to the underlying microscopic-particle dynamics. While such an approach has already been taken earlier in relation to the deformation gradient [
31], it is still lacking for the distortion field, to the best of our knowledge. Linking a particle-based description to continuum quantities is a prototypical example of coarse graining and multiscale modeling [
32,
33]. Doing so is not only beneficial for a fundamental understanding of the continuum model, it also helps to devise efficient hierarchical modeling strategies.
The developments in this paper concern the deformation in
D-dimensional Euclidean space, e.g.,
\(D=3\) for bulk materials or
\(D=2\) for plane, flat domains. It is pointed out for completeness that one is not dealing here with curved spaces, e.g., a curved surface such as the surface of a soap bubble, because there the dynamics involves contributions that are perpendicular to the tangent spaces to the surface [
34].
Before we proceed, a word on notation: Discrete particles are enumerated by Greek indices (
\(\alpha \),
\(\beta \),...), while coordinates are labeled with Latin indices: lowercase for the current configuration (
i,
j,...) and uppercase for the natural configuration (
I,
J,...). For example, coordinates of particle
\(\alpha \) in the current and natural configurations are given by
\(r_{\alpha }^{i}\) and
\(R_{\alpha }^{I}\), respectively, while
\(A{}^{I}{}_{\! j}\) denotes a component of the distortion tensor. In all that follows, the notation of tensor calculus according to Ricci-Curbastro and Levi-Civita [
35,
36] is used. This implies that upper and lower indices are distinguished corresponding to contravariant and covariant components of tensors. Throughout this paper, the upper/lower positioning of indices is used, and it is always understood—without explicit mention–that one can transition between the two representations by multiplication with the appropriate metric. Summations over particle indices are always spelled out, while for summation over coordinates the Einstein summation convention is used, and summations are only permitted over index pairs of which one index is upper and the other is lower. For practical convenience, for the developments and derivations in this paper, Cartesian coordinates with Euclidean metric–given by the Kronecker delta (where the distinction between upper and lower indices is irrelevant)–are used, without loss of generality. However, specifically in Sect.
6 the main results of the paper are reformulated for generalized (curvilinear) coordinates. Finally, for functions
f of macroscopic position, the short-hand notation
\(f \equiv f(\textbf{r})\) and
\(f^\prime \equiv f(\textbf{r}^\prime )\) is often used for better readability.
The paper is organized as follows: Sect.
2 starts with deriving a best-fit expression for the distortion field in terms of discrete particle positions on a finer level, which is then employed to examine the reversible (affine) and irreversible (plastic) dynamics of the distortion by direct calculation. In Sect.
3, the General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework is introduced, with special emphasis on its systematic procedure for statistical-mechanics-based coarse graining. This procedure is then employed to examine in more detail the reversible dynamics (Sect.
4) and the irreversible dynamics (Sect.
5) of the distortion, respectively. In Sect.
6, the main results of this paper are presented in covariant form, for generalized coordinates. Conclusions are drawn in Sect.
7. The main achievements are summarized as “Result” in the respective sections.
3 Method: nonequilibrium thermodynamics and statistical mechanics
To approach the dynamics of the distortion from a wider perspective, we make use of nonequilibrium thermodynamics. While a multitude of modeling frameworks have been developed in this field over the years, we specifically choose the General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework [
47‐
49]; the main reason for this choice is that this framework also comes with a procedure for coarse graining, i.e., for establishing links between models on different levels of description, which is the main focus of this paper in relation to the distortion. In the following, only a very concise summary of the framework is outlined as needed for this paper; for more details, the reader is referred to the original papers on GENERIC [
47,
48] and its coarse graining procedure [
50], the comprehensive books [
49,
51], or applications of this framework to the case of fluid and solid mechanics, e.g., [
30,
31,
52].
Let
\(\mathcal {X}\) denote the set of fundamental variables, for which a closed set of evolution equations is sought; with reference to the specific case of interest in this paper,
\(\mathcal {X}\) could contain the momentum density, the mass density, the temperature (or density of internal energy or entropy), and the distortion. According to GENERIC, the evolution equations are given by [
47‐
49]
$$\begin{aligned} d\mathcal {X}= \mathcal {L}\cdot \frac{\delta E}{\delta \mathcal {X}} dt + \mathcal {M}\cdot \frac{\delta S}{\delta \mathcal {X}} dt \, , \end{aligned}$$
(9)
with total energy
E, entropy
S, Poisson operator
\(\mathcal {L}\), and friction operator
\(\mathcal {M}\). The two contributions on the r.h.s. of Eq. (
9) are reversible (e.g., affine deformation, elasticity) and irreversible (e.g., plasticity) in nature, respectively. The dot (
\(\cdot \)) does not only represent summation over discrete indices in the sense of different elements of
\(\mathcal {X}\), but may also stand for integration over continuous labels in the case of field theories, e.g., as what will be discussed in the remainder of this paper.
The two operators must satisfy various properties: On the one hand, (L1) \(\mathcal {L}\) must be anti-symmetric, (L2) \(\mathcal {L}\) must satisfy the Jacobi identity (to ensure time-structure invariance), and (L3) the derivative of entropy, \(\delta S/\delta \mathcal {X}\), must be in the null-space of \(\mathcal {L}\), i.e., \(\mathcal {L}\cdot (\delta S/\delta \mathcal {X}) = 0\); on the other hand, (M1) \(\mathcal {M}\) must be Onsager–Casimir symmetric, (M2) \(\mathcal {M}\) must be positive semi-definite, and (M3) the derivative of energy, \(\delta E/\delta \mathcal {X}\), must be in the null-space of \(\mathcal {M}\), i.e., \(\mathcal {M}\cdot (\delta E/\delta \mathcal {X}) = 0\). It can be shown that these six conditions imply the conservation of energy E and that the entropy S is a non-decreasing function of time.
The fluctuation–dissipation theorem [
53‐
55] states that, whenever there is dissipation, there are in principle also fluctuations present; whether the fluctuations are practically relevant is case specific. In GENERIC, the dissipation originates from the presence of the friction operator
\(\mathcal {M}\). Incorporation of (Wiener process based) fluctuations in a thermodynamically consistent manner extends the GENERIC from Eq. (
9) to the following stochastic differential equation (SDE) [
47,
49,
50], using the Itô interpretation of stochastic calculus [
56,
57],
$$\begin{aligned} d\mathcal {X}= \mathcal {L}\cdot \frac{\delta E}{\delta \mathcal {X}} dt + \mathcal {M}\cdot \frac{\delta S}{\delta \mathcal {X}} dt + k_\mathrm{{B}}\frac{\delta }{\delta \mathcal {X}}\cdot \mathcal {M}dt + \mathcal {B}\cdot d\mathcal {W}\, , \end{aligned}$$
(10)
with Boltzmann constant
\(k_\mathrm{{B}}\) and where
\(d\mathcal {W}\) stands for multicomponent white noise, more precisely, for the increment of a multicomponent Wiener process [
56,
57]. The “strength”
\(\mathcal {B}\) of the fluctuating contribution in Eq. (
10) is related to the friction operator by way of
$$\begin{aligned} \mathcal {B}\cdot \mathcal {B}^{\textrm{T}} = 2 k_\textrm{B} \mathcal {M}\, , \end{aligned}$$
(11)
which ensures that the fluctuation–dissipation theorem is respected. The deterministic case Eq. (
9) can be recovered from the fluctuating one, Eq. (
10), in the limit
\(k_\textrm{B} \rightarrow 0\), while leaving the building blocks
E,
S,
\(\mathcal {L}\), and
\(\mathcal {M}\) unchanged.
As already mentioned, the GENERIC framework can be used also for establishing a relation between different levels of description. Imagine to coarse grain from a fine level with variables
\(\mathcal {X}_1\) to a coarser level with variables
\(\mathcal {X}_2\). The instantaneous value of the level 2 quantities,
\(\Pi _2\), can be expressed in terms of level 1 quantities,
\(\Pi _2 = \Pi _2(\mathcal {X}_1)\), and the coarse-grained variables are eventually given by
\(\mathcal {X}_2 = \langle \Pi _2\rangle \), where
\(\langle \ldots \rangle \) denotes the average with an appropriate distribution function [
49,
50]. In analogy to conventional statistical mechanics, one can derive expressions for the energy and entropy,
\(E_2\) and
\(S_2\), on level 2 based on level 1 information; since this is not pursued in this paper, it is not presented in this concise summary, and we refer to [
49,
50] for details on this specific aspect.
The reversible and irreversible dynamics on level 1 is also represented on level 2. Concretely, this is reflected by the relations [
31,
49,
50]
$$\begin{aligned} \mathcal {L}_2= & {} \left\langle \frac{\delta \Pi _2}{\delta \mathcal {X}_1} \cdot \mathcal {L}_{1} \cdot \frac{\delta \Pi _2}{\delta \mathcal {X}_1} \right\rangle , \end{aligned}$$
(12)
$$\begin{aligned} \mathcal {M}^\prime _{2}= & {} \left\langle \frac{\delta \Pi _{2}}{\delta \mathcal {X}_{1}} \cdot \mathcal {M}_{1} \cdot \frac{\delta \Pi _{2}}{\delta \mathcal {X}_{1}} \right\rangle , \end{aligned}$$
(13)
where the contractions (
\(\cdot \)) work between the level 1 operator and
\(\mathcal {X}_1\). Furthermore, there might be an additional dissipative contribution on level 2, since some features of the dynamics that were explicitly resolved on level 1 may be seen on the coarser (and slower) level 2 as rapid, uncontrollable noise. In this case, there is a second contribution to the friction operator on level 2 [
31,
49,
50],
$$\begin{aligned} \mathcal {M}^{\prime \prime }_2 = \frac{1}{k_\mathrm{{B}}} \int ^\tau _0 \left\langle \dot{\Pi }^{\textrm{f}}_2(t+s) \dot{\Pi }^{\textrm{f}}_2(t) \right\rangle ds, \end{aligned}$$
(14a)
which involves the correlation of the fluctuating contribution to the dynamics of
\(\Pi _2\),
\(\dot{\Pi }^{\textrm{f}}_2\), at different instances in time. The quantity
\(\tau \) denotes the separating time-scale between levels 1 and 2. If the correlation function decays sufficiently fast, Eq. (
14a) can be written in the form [
49]
$$\begin{aligned} \left[ 1+\varepsilon (\mathcal {X}_2)\varepsilon (\mathcal {X}_2)\right] \mathcal {M}^{\prime \prime }_2 = \frac{1}{k_\mathrm{{B}}\tau } \left\langle \Delta _\tau \Pi ^{{\textrm{f}}}_2 \Delta _\tau \Pi ^{\textrm{f}}_2 \right\rangle , \end{aligned}$$
(14b)
where
\(\Delta _\tau \Pi ^{\textrm{f}}_2\) stands for the fluctuation-related increment of
\(\Pi _2\) in the time interval
\(\tau \). The prefactor on the left-hand side (l.h.s.) of Eq. (
14b) is a short-hand notation for the following: Since
\(\mathcal {X}_2\) in general contains several variables,
\(\mathcal {M}_2\) has a matrix structure. The prefactor on the l.h.s. in Eq. (
14b) means that each element of
\(\mathcal {M}_2\) is to be multiplied by a factor that depends on the parity under time reversal (
\(\varepsilon = \pm 1\)) of the two respective variables concerned, which is directly related to the Onsager–Casimir reciprocal relations [
58‐
60]. In particular, if the two variables concerned have the same parity, the prefactor equals 2, which is the case for what will be discussed in this paper.
A slightly different formulation of GENERIC, where the friction operator is replaced by a dissipation potential [
47], is advocated in [
61]. Although the formulation with a friction operator is more general than the formulation with dissipation potential [
62] (for instance, because the operator may not always be symmetric), the dissipation potential is appealing because of an intimate connection with the principle of large deviations [
63,
64]. It remains unclear, however, which of the formulations should be preferred in general. Here, we choose the formulation with the friction operator because it contains a standardized procedure of statistical-mechanics-type coarse graining [
49,
50].
The following two sections focus on the derivation of both the reversible and irreversible parts of the dynamics of the distortion from the level of discrete microscopic particles.
While the procedure in the previous sections has used Cartesian coordinates, the goal of this section is to provide guidelines on how to express the main results in generalized (curvilinear) coordinates, namely the distortion Eq. (
3), the reversible contribution to the evolution equations Eq. (
22) with stress tensor Eq. (
23) (and Eq. (
24)), the irreversible contributions Eq. (
40) with the Mandel stress Eq. (
41) and the plastic deformation-rate tensor in the natural configuration Eq. (
42), as well as the discussion about incompressibility in Sec.
5.3. There are four points that need to be discussed w.r.t. the use of generalized coordinates.
Point 1: Wherever Kronecker deltas appear with both indices up or both indices down, respectively, they need to be replaced by the appropriate metric:
\(\delta ^{ij} \rightarrow g^{ij}\),
\(\delta _{ij} \rightarrow g_{ij}\),
\(\delta ^{IJ} \rightarrow G^{IJ}\), and
\(\delta _{IJ} \rightarrow G_{IJ}\), where the so-called Eulerian metric
\(\textbf{g}\) and the so-called material metric
\(\textbf{G}\) denote the metrics in the current and natural configuration, respectively. Furthermore, these metrics are also to be used whenever indices are raised or lowered in the respective configuration. While the presence of the Eulerian metric helps to preserve the covariance of the equations, the material metric is necessary for instance to calculate the energy. The two (Eulerian and material) metrics act on different manifolds, which means that they are different in general [
28]. If the manifolds have neither curvature nor torsion, using the Levi-Civita connection [
82], we can choose such coordinate systems that the metrics are constant (by Frobenius theorem [
83]) and equal to the unit matrices [
84]. The material metric has already been introduced in [
85] for modeling of pre-stressed materials and in [
86] for space–time compatibility reasons.
Point 2: Partial derivatives w.r.t. a spatial coordinate,
\(\partial _i \equiv \partial / \partial r^i\), need to be expressed as covariant derivatives, involving the Christoffel symbols, see [
35,
36,
87] in general and also [
11] in the context of mechanics. Obviously, the covariant derivative reduces to the partial derivative in the case of Cartesian coordinates.
Point 3: Even if working in generalized coordinates, it is recommended to interpret densities w.r.t. the unit volume element in the Cartesian setting. However, this implies that, when integrating the density for obtaining the corresponding extensive quantity, the (square root of the determinant of the) metric needs to be taken into account.
Point 4: It may be sufficient to keep the definition of the distortion Eq. (
3) in the Cartesian formulation, for the following reason. Due to the localization function, the region of interest is small on a macroscopic scale; in this region of interest, a local Cartesian coordinate system can be introduced as a reasonable approximation.
It is noted that particularly for Point 1 and Point 2 it proves beneficial that systematic use has been made of the notation of tensor calculus according to Ricci-Curbastro and Levi-Civita [
35,
36] throughout this paper and that Kronecker deltas have been used with appropriate positioning of indices.
7 Conclusion
In this paper, we have introduced a particle-based definition of the Eulerian distortion field by using a best-fit procedure in a certain zone-of-influence, where the latter can be controlled by a localization function (see Sect.
2). Specifically, the best-fit distortion Eq. (
3) is defined as the minimizer of the error Eq. (
2) that measures the discrepancy between distances in the natural and current configurations, respectively.
While a simplified analysis of the dynamics of the distortion is pursued in Sect.
2 by direct calculation, there is a need for a more thorough analysis in the form of systematic coarse graining from the microscopic dynamics, in particular to shed more light on the irreversible dynamics of the distortion. In other words, the simplified point of view of Sect.
2 has been made more precise by using the GENERIC framework and its procedure for coarse graining, as summarized in Sect.
3. Within GENERIC, evolution equations are composed of a reversible (Hamiltonian) part and an irreversible (dissipative) part, where all relevant building blocks can be expressed in terms of liner-level information. In this way, the evolution equations for the distortion and the densities of mass, momentum, and entropy have been derived by coarse graining in Sect.
4 and Sect.
5. With regard to the reversible contribution to the evolution equations (Sect.
4), the underlying Poisson operator Eq. (
21) has been derived from the canonical Poisson operator for classical particles.
In Sect.
5, the emergence of the irreversible dynamics of the distortion has been examined in detail. Specifically, the relaxation of the distortion—representative of inelastic/plastic deformation—has been linked to the underlying dynamics of the constituent particles, Eq. (
29), which is expressed by the friction operator Eq. (
28) with Eqs. (
33) and (
34). Irrespective of the material considered, it has been established that the Mandel stress is the relevant driving force for the relaxation of the distortion. The coarse graining procedure has been illustrated w.r.t. the irreversible dynamics with the help of two examples, an amorphous isotropic solid and a crystal with one slip system. Finally, it has been examined how the incompressibility of distortion relaxation is expressed not only on the coarse-grained continuum level, but also on the fine level of the constituent particles.
In the following, four perspectives for future work are identified and discussed briefly.
Outlook 1: The results of this paper can be used not only for analyzing the results of particle-based simulations (e.g., molecular dynamics simulations of atoms or of slightly coarse-grained, so-called united, atoms) in terms of extracting the field of distortion as such, but also for studying the rapid contributions to the fine-scale dynamics for eventually obtaining the friction operator (see Sect.
5 for details). In this respect, it is crucial to examine whether the fluctuations can be represented adequately in terms of stochastic differential equations (SDE) as discussed in Sect.
5, or whether there are rather rare but large events that prevail [
88], where the latter would require other means of coarse graining (e.g., see [
72,
73]). An equally important question is the one of time-scale separation between the different levels of description; a lack of time-scale separation (as observed, e.g., in dislocation systems [
89]) complicates the coarse graining significantly.
Outlook 2: In order to account for substantial inhomogeneities of the deformation, it has been proposed to let the strain energy depend not only on a certain measure of deformation itself, but also on its gradient, see, e.g., [
90]. If the measure of deformation is given by the Eulerian distortion, this implies that the strain energy depends not only on the distortion
\(A{}^{I}{}_{\! j}\), but also on its partial derivative w.r.t. the spatial coordinates,
$$\begin{aligned} H{}^{I}{}_{\! j k}= & {} \partial _k A{}^{I}{}_{\! j}. \end{aligned}$$
(53)
In principle,
\(H{}^{I}{}_{\! j k}\) could be calculated as the partial derivative of the distortion
\(A{}^{I}{}_{\! j}\), once the latter has been determined according to Sect.
2.1. However, particularly when discrete-to-continuum relations are of interest [
90], it may be preferable to obtain
\(H{}^{I}{}_{\! j k}\) directly from the arrangement of the particles, in analogy to Eq. (
3) for the distortion
\(A{}^{I}{}_{\! j}\). This can actually be achieved by following the procedures developed in [
41] or in [
91,
92], as detailed in “Appendix D.” While following [
41] leads to an expression for
\(H{}^{I}{}_{\! j k}\) that is inherently symmetric,
\(H{}^{I}{}_{\! k j} = H{}^{I}{}_{\! j k}\), the approach taken in [
91,
92] is more general, see “Appendix D” for details. In either case, it might be interesting to extend the results for the dynamics of
\(A{}^{I}{}_{\! j}\) derived in this paper to the case where both
\(A{}^{I}{}_{\! j}\) and
\(H{}^{I}{}_{\! j k}\) (as derived in “Appendix D”) are independent dynamic variables.
Outlook 3: It is mentioned in Sect.
1 that in the presence of irreversible deformation, the distortion is in general not compatible, i.e., not curl-free. In the case of dislocations, the curl of the distortion is related to the dislocation density tensor [
21,
24‐
27] and the Burgers tensor [
16,
46]. Such a relation has actually been employed successfully in [
91,
92] for
detecting dislocations based on configurations of the discrete particles, i.e., atoms. One may wonder whether even for a wider class of systems, the curl of the distortion is a useful tool for detecting carriers of plastic deformation. With Eq. (
53), the curl of the distortion (expressed in Cartesian coordinates) reads
$$\begin{aligned} \left( \text {curl}\textbf{A}\right) ^{Ii} = \varepsilon ^{ijk} H{}^{I}{}_{\! k j}, \end{aligned}$$
(54)
with Levi-Civita’s permutation symbol
\(\varepsilon ^{ijk}\) (taking values
\(-1\), 0, 1); the extension to generalized coordinates requires simply to introduce the prefactor
\(1/\sqrt{g}\) on the r.h.s. of Eq. (
54) [
36,
93], where
g denotes the determinant of the metric in the current configuration. In line with Outlook 2, there are two possibilities for the application of Eq. (
54) on the level of the particles: Either the spatial derivatives of the distortion are calculated after the distortion
\(A{}^{I}{}_{\! j}\) has been obtained by the best-fit procedure in Sect.
2.1, or one makes use of
\(H{}^{I}{}_{\! j k}\) that itself is obtained directly by way of a best-fit procedure (as described in “Appendix D”). In the latter case, the approach following [
41] results in a vanishing curl (due to
\(H{}^{I}{}_{\! k j} = H{}^{I}{}_{\! j k}\)) and is therefore insufficient for the discussion of incompatibility. Instead, the approach following [
91,
92] should be used. However, while [
91,
92] propose a two-step (i.e., sequential) procedure for the calculation of the best-fit measure of deformation and its derivative, it is worth pursuing options for a one-step (i.e., simultaneous) procedure, as mentioned briefly in the discussion section of [
92]. Whatever procedure is used for the determination of the (non-vanishing) curl of the distortion, it is emphasized that no
a priori assumptions about the defect structure enter that determination; rather, it processes directly the arrangements of particles in the current and natural configurations. If the so-obtained mapping between these two configurations turns out to be incompatible, one may
use the curl of the distortion as a tool for
detecting defects, particularly dislocations, along the lines of [
91,
92].
Outlook 4: As mentioned in Sect.
1, this paper deals with flat spaces. In contrast, when studying spaces with curvature, additional care needs to be taken and amendments to the procedure developed in this paper may be required. This is because one cannot find a parametrization such that the components of the metric are equal to the Kronecker delta everywhere.