# Interface-dominated plasticity and kink bands in metallic nanolaminates

Abhishek Arora Rajat Arora Amit Acharya Department of Civil & Environmental
Engineering, Carnegie Mellon University, Pittsburgh, PA 15213Advanced Micro
Devices, Austin, TX 78735Department of Civil & Environmental Engineering, and
Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA
15213, email<EMAIL_ADDRESS>
###### Abstract

The theoretical and computational framework of finite deformation mesoscale
field dislocation mechanics (MFDM) is used to understand the salient aspects
of kink-band formation in Cu-Nb nano-metallic laminates (NMLs). A conceptually
minimal, plane-strain idealization of the three-dimensional geometry,
including crystalline orientation, of additively manufactured NML is used to
model NMLs. Importantly, the natural jump/interface condition of MFDM imposing
continuity of (certain components) of plastic strain rates across interfaces
allows theory-driven ‘communication’ of plastic flow across the laminate
boundaries in our finite element implementation. Kink bands under layer
parallel compression of NMLs in accord with experimental observations arise in
our numerical simulations. The possible mechanisms for the formation and
orientation of kink bands are discussed, within the scope of our idealized
framework. We also report results corresponding to various parametric studies
that provide preliminary insights and clear questions for future work on
understanding the intricate underlying mechanisms for the formation of kink
bands.

##  1 Introduction

A fundamental challenge in the development of efficient engineering materials
is to design a material that has both high strength and high toughness, as
most materials have exclusively either of these properties. Kinking in nano-
metallic laminates is a deformation mechanism that may circumvent this
challenge, as it leads to enhanced plasticity (ductility) at an almost
constant level of stress after yielding [12]. Kink bands, in the present
context of a material with an intrinsic layered structure, refer to a thin
band of localized deformation that is transverse - and most commonly not
normal - to the intrinsic layering, as shown in Fig. 1 (a). Shear band
formation has been the predominantly observed form of localization in NMLs as
reported in [29, 13]. In these studies, NMLs are compressed perpendicular to
the layer direction in micropillar configurations, meaning the compression
loading direction is parallel to the layer normal direction. While this is a
somewhat physically intuitive mode of deformation localization through
rotation of the layers towards the loading axis and slipping along softer
layers, kink band formation in NMLs for sufficiently small layer thickness
under layer parallel compression has also been demonstrated [32]. In layer
parallel compression of NML, the compression loading direction is
perpendicular to the layer normal direction as shown in Fig. 1 (a). The
formation of such kink bands and its dependence on the size of layers is quite
non-intuitive, and it is the study of this phenomenon that forms the subject
of this paper; the modeling of the development of shear bands in NMLs in
micropillar configurations under layer perpendicular compression is a subject
worthy of study on its own merits, a challenge that has not been successfully
solved as yet [46].

In general, the formation of shear or kink bands, due to strain localization,
is detrimental as it produces sites of narrow planes under intense shear or
lattice rotations, which acts as a source for crack initiation and leads to
eventual material failure. For instance, it was reported in [25] that for
grain sizes smaller than 300 nm in iron (Fe), shear bands are formed leading
to strain-softening and failure at low values of plastic strain. Similarly,
graphite-fiber-reinforced epoxy composites fail under compression loading due
to the formation of kink bands [33]. However, for layer parallel compression
of NMLs with sufficiently small laminate widths, a large amount of plastic
deformation is observed before failure without much reduction in the strength
of the material in the experimental work of Thomas et al. [32] in Cu-Nb
nanolaminates. It was reported there that the samples with layer thickness
greater than 250 nm deform homogeneously, while the deformation behavior of
samples with thicknesses 65 or 30 nm was inhomogeneous, and enhanced plastic
deformation was observed after yield at a more-or-less constant load [32].
Recently, Zhang et al. [47] investigated the kink band formation process in
Cu/Nb NMLs via in situ micropillar compression along the layer-parallel
direction in a scanning electron microscope (SEM). It was reported that after
the onset of kinking, dislocations active on the slip systems nearly parallel
to the interface in NMLs are responsible for the band evolution. Cahn [15]
showed the formation of kink bands in aluminum single crystals, and it was
suggested that these bands are composed of dipolar arrays of edge dislocations
aligned perpendicular to the primary slip direction.

(a)

(b)

Figure 1: (a) Schematic of an idealized kink band in layer parallel
compression of NML, (b) geometry of the kink band at $|e_{N}|=25\%$
engineering strain in the experimental work of compression of Cu/Nb NMLs.
(Figure in (b) reprinted from [47] with permission from Elsevier.)

With respect to modeling, Zecevic et al. [46] used Gurtin’s [24] strain
gradient crystal plasticity model augmented with various choices of the defect
energy function within a large strain elasto-viscoplastic Fast Fourier
transform-based computational technique to model the compression of Cu/Nb
NMLs. Arora and Acharya [1] modeled a 2-D single crystal under simple shearing
using the mesoscale field dislocation mechanics (MFDM) framework in a finite
deformation setting. In a special single slip study, kink bands were observed
in the form of dipolar dislocation walls perpendicular to the single slip
plane [1]. Kink band formation in a polycrystalline Mg alloy under compression
was computationally studied using a strain-gradient crystal plasticity
framework [44]. Asaro and Rice [8] described the onset of a shear and kink
band from a bifurcation analysis of the classical, finite deformation, single
slip constitutive behavior. Forest [21] used small deformation generalized
crystal plasticity (Cosserat theory of crystal plasticity) to separate
bifurcation modes (slip and kink bands), i.e. kink bands occur later than slip
bands, and this is unlike the prediction of classical plasticity.

In this work, we model the compression of composite nanolaminates within a
plane strain idealization, using the theoretical and computational framework
of MFDM. The interface conditions in MFDM impose continuity of certain
components of the total plastic strain rate, which models continuous plastic
flow across the laminate interfaces. We consider a ‘minimal’ model setup in
considering 3 co-planar slip systems for each material in the bi-metallic
laminate structure and show that kink bands are formed in qualitative accord
with the experimental observations of [32, 47]. We also report on other
studies within this idealized setup and discuss whether kink bands form or not
for different cases.

The outline of the paper is as follows: after presenting some notational
details used in this paper in the following paragraph, Section 2 provides a
brief review of the theory of MFDM. For more details on the theoretical and
computational framework of MFDM, the interested reader is referred to [1, 11,
2, 10]. Section 3 discusses the various successes of the MFDM framework with
correspondence with experimental studies in many instances. Section 4 shows
the numerical results of kink-band formation in composite nanolaminates within
our idealization using the computational framework of MFDM. We also discuss
various parametric studies of our model characterizing this complicated
problem of kink band formations in NMLs. Section 5 presents some concluding
remarks and the scope of future work for modeling compression of NMLs using
our framework.

Vectors and tensors are represented by boldface lower and upper-case letters,
respectively. The action of a second-order tensor ${\mathbold A}$ on a vector
${\mathbold b}$ is denoted by ${\mathbold A}{\mathbold b}$. The inner product
of two vectors is denoted by ${\mathbold a}\cdot{\mathbold b}$, while the
inner product of two second-order tensors is denoted by ${\mathbold
A}:{\mathbold B}$. A rectangular Cartesian coordinate system is invoked for
the ambient space and all vector and tensor components are expressed with
respect to the basis of this coordinate system. The partial derivative of the
quantity $(\cdot)$ w.r.t. the $x_{i}$ coordinate of this coordinate system is
denoted by $(\cdot)_{,i}$, and ${\mathbold e}_{i}$ denotes the unit vector in
the $x_{i}$ direction. The time derivative of a quantity $(\cdot)$ is denoted
by $\dot{(\cdot)}$. Einstein’s summation convention is always implied unless
mentioned otherwise. The symbols $\mathop{\rm grad}\nolimits$, $\mathop{\rm
div}\nolimits$, and $\mathop{\rm curl}\nolimits$ denote the gradient,
divergence, and curl on the current configuration, respectively. For a second
order tensor ${\mathbold A}$, vectors ${\mathbold v}$, ${\mathbold a}$, and
${\mathbold c}$, and a spatially constant vector field ${\mathbold b}$, the
operations of $\mathop{\rm div}\nolimits$, $\mathop{\rm curl}\nolimits$, and
the cross-product of a tensor $(\times)$ with a vector are defined as follows:

$\displaystyle\left(\mathop{\rm div}\nolimits{\mathbold
A}\right)\cdot{\mathbold b}=\mathop{\rm div}\nolimits\left({\mathbold
A}^{T}{\mathbold b}\right),$ $\displaystyle\qquad\forall\>\>{\mathbold b}$
$\displaystyle{\mathbold b}\cdot\left(\mathop{\rm curl}\nolimits{\mathbold
A}\right){\mathbold c}=\left[\mathop{\rm curl}\nolimits\left({\mathbold
A}^{T}{\mathbold b}\right)\right]\cdot{\mathbold c},$
$\displaystyle\qquad\forall\>\>{\mathbold b},{\mathbold c}$
$\displaystyle{\mathbold c}\cdot\left({\mathbold A}\times{\mathbold
v}\right){\mathbold a}=\left[\left({\mathbold A}^{T}{\mathbold
c}\right)\times{\mathbold v}\right]\cdot{\mathbold a},$
$\displaystyle\qquad\forall\>\>{\mathbold a},{\mathbold c}.$

In rectangular Cartesian coordinates, these are denoted by

$\displaystyle\left(\mathop{\rm div}\nolimits{\mathbold
A}\right)_{i}=A_{ij,j},\quad\left(\mathop{\rm curl}\nolimits{\mathbold
A}\right)_{ri}=\varepsilon_{ijk}A_{rk,j},\quad\left({\mathbold
A}\times{\mathbold v}\right)_{ri}=\varepsilon_{ijk}A_{rj}v_{k},$

where $\varepsilon_{ijk}$ are the components of the third order alternating
tensor ${\mathbold X}$.

When we refer to different slip systems with respect to their slip vector
orientation, we mean that those orientations were in the initial unstretched
lattice, and do not represent the orientations of the corresponding slip
vectors in the deformed lattice, unless mentioned otherwise.

##  2 Review of MFDM theory

The governing equations for MFDM [11] are as follows:

$\displaystyle\mathop{\rm curl}\nolimits\boldsymbol{W}=-\boldsymbol{\alpha},$
(3a) $\displaystyle\dot{{\mathbold W}}+{\mathbold W}{\mathbold
L}=\boldsymbol{\alpha}\times{\mathbold V},$ (3b) $\displaystyle\mathop{\rm
div}\nolimits\left[{\mathbold T}({\mathbold
W})\right]=\bigg{\\{}\begin{array}[]{l}\boldsymbol{0},\quad\text{quasistatic}\\\
\rho\dot{{\mathbold v}},\quad\text{dynamic}.\end{array}$ (3e)

In Eq. (3), $\boldsymbol{W}$ is the inverse of the elastic distortion tensor
($\boldsymbol{F}^{e}$), $\mathbold{\alpha}$ denotes the dislocation density
tensor (Nye’s tensor), ${\mathbold v}$ denotes the material velocity,
${\mathbold L}=\mathop{\rm grad}\nolimits{\mathbold v}$ is the velocity
gradient, ${\mathbold T}$ is the (symmetric) Cauchy stress tensor, and
${\mathbold V}$ denotes the dislocation velocity field with respect to the
material. Eq. (3a) is the incompatibility equation for the inverse elastic
distortion tensor, while Eq. (3b) is derived from the statement of
conservation of the Burgers vector content of an arbitrary area patch in the
current configuration [7]. Eq. (3e) is the linear momentum balance for both
quasistatic and dynamic cases.

Eq. (3b) is a system of coupled Hamilton-Jacobi equations for which, to our
knowledge, no rigorous mathematical theory or numerical schemes exist in the
literature. To computationally approximate the governing equations, a Stokes-
Helmholtz decomposition of the field ${\mathbold W}$ is introduced, expressing
it as a sum of a gradient $(\mathop{\rm grad}\nolimits{\mathbold f})$ and a
divergence-free part ($\boldsymbol{\chi}$) [1]. This decomposition replaces
the tensor field ${\mathbold W}$ with a tensor and a vector field in our
computations, but it is relatively easier to develop numerical schemes for
this larger set of governing equations [11].

The extended set of governing equations for MFDM [11] are shown below:

$\displaystyle\mathring{\boldsymbol{\alpha}}\equiv(\mathop{\rm
div}\nolimits(\boldsymbol{v}))\boldsymbol{\alpha}+\dot{\boldsymbol{\alpha}}-\boldsymbol{\alpha}{\mathbold
L}^{T}=-\mathop{\rm curl}\nolimits(\boldsymbol{\alpha}\times{\mathbold
V}+{\mathbold L}^{p}),$ (4a) $\displaystyle\begin{array}[]{c}{\mathbold
W}=\boldsymbol{\chi}+\mathop{\rm grad}\nolimits{\mathbold
f},\>\boldsymbol{F}^{e}=\boldsymbol{W}^{-1},\\\ \mathop{\rm
curl}\nolimits\boldsymbol{\chi}=-\boldsymbol{\alpha},\\\ \mathop{\rm
div}\nolimits\boldsymbol{\chi}=\boldsymbol{0},\end{array}\Bigg{\\}}$ (4e)
$\displaystyle\mathop{\rm div}\nolimits\left(\mathop{\rm
grad}\nolimits\dot{\boldsymbol{f}}\right)=\mathop{\rm
div}\nolimits\left(\boldsymbol{\alpha}\times\boldsymbol{V}+{\mathbold
L}^{p}-\dot{\boldsymbol{\chi}}-\boldsymbol{\chi}\boldsymbol{L}\right),$ (4f)
$\displaystyle\mathop{\rm div}\nolimits\left[{\mathbold T}({\mathbold
W})\right]=\bigg{\\{}\begin{array}[]{l}\boldsymbol{0},\quad\text{quasistatic}\\\
\rho\dot{{\mathbold v}},\quad\text{dynamic}.\end{array}$ (4i)

Here, (4a) is the evolution equation for the dislocation density tensor, and
(4e) is the incompatibility equation in terms of $\boldsymbol{\chi}$ with a
condition that $\mathop{\rm div}\nolimits\boldsymbol{\chi}=\boldsymbol{0}$,
while ${\mathbold f}$ denotes the plastic position vector, and the evolution
equation for ${\mathbold f}$ is given by (4f). ${\mathbold L}^{p}$ is the
additional mesoscale field which represents the averaged rate of plastic
straining due to all dislocations that cannot be represented by
$\mathbold{\alpha}\times{\mathbold V}$, where both fields in the product
represent space-time running averages of the corresponding microscopic fields.
This is due to the fact that the average of products is not the same as the
product of averages, in general. Hence, all fields in the MFDM framework are
running space-time averages of the corresponding fields of the field
dislocation mechanics (FDM) theory; except for ${\mathbold L}^{p}$, as there
is no corresponding field in FDM theory to ${\mathbold L}^{p}$.

To close the above set of equations in (4), constitutive statements for
$\boldsymbol{V},\boldsymbol{L}^{p},\boldsymbol{T}$ are used, consistent with
the mechanical dissipation being non-negative [1].

Initial conditions for $\mathbold{\alpha},{\mathbold f}$ and boundary
conditions for $\mathbold{\alpha},{\mathbold f},\mathbold{\chi}$, and
${\mathbold v}$ are specified, for a well-set evolution in MFDM.

###  2.1 Boundary conditions

  * •

The $\boldsymbol{\alpha}$ evolution equation (shown in Eq. (4a)) has a
convective boundary condition of the form $(\mathbold{\alpha}\times{\mathbold
V}+{\mathbold L}^{p})\times{\mathbold n}=\boldsymbol{\Phi}$, where
$\boldsymbol{\Phi}$ is a second order tensor valued function of position and
time on the boundary, characterizing the flux of dislocations at the surface
satisfying the constraint $\boldsymbol{\Phi}{\mathbold n}=\mathbf{0}$. Here,
${\mathbold n}$ isThe outward unit normal field on the boundary.

The boundary condition is specified in two ways: (a) Plastically constrained:
$\boldsymbol{\Phi}({\mathbold x},t)=\mathbf{0}$ is specified at a point
${\mathbold x}$ on the boundary for all times, which ensures that there is no
outflow of dislocations at that point of the boundary, and only parallel
motion along the boundary is allowed. (b) Plastically unconstrained: A less
restrictive boundary condition where $\hat{{\mathbold L}}^{p}\times{\mathbold
n}$ is simply evaluated at the boundary (akin to an outflow condition), along
with the specification of dislocation flux $\mathbold{\alpha}({\mathbold
V}\cdot{\mathbold n})$ on the inflow part of the boundary. Additionally, for
all calculations presented in this paper $(\mathop{\rm
curl}\nolimits\mathbold{\alpha}\times{\mathbold n})=\bf 0$ is imposed, a
particular specification of a boundary condition that arises from simple
mathematical modeling of the manifestation of dislocation core energy at the
mesoscale.

  * •

For the incompatibility equation, $\boldsymbol{\chi}\boldsymbol{n}=\mathbf{0}$
is applied on the outer boundary of the domain, which along with the system
(4e) ensures that $\mathbold{\chi}$ vanishes when $\mathbold{\alpha}$ is zero
in the entire domain.

  * •

The $\boldsymbol{f}$ evolution equation requires a Neumann boundary condition
i.e., $(\mathop{\rm grad}\nolimits\dot{{\mathbold f}}){\mathbold
n}=(\mathbold{\alpha}\times{\mathbold V}+{\mathbold
L}^{p}-\dot{\mathbold{\chi}}-\mathbold{\chi}{\mathbold L}){\mathbold n}$ on
the outer boundary of the domain.

  * •

The material velocity boundary conditions are applied based on the loading
type, which is discussed later in Section 4.

###  2.2 Initial conditions

  * •

The initial condition $\boldsymbol{\alpha}(\boldsymbol{x},0)=\boldsymbol{0}$
is assumed for all simulations here.

  * •

In general, the initial condition for $\boldsymbol{f}$ is obtained by solving
for $\mathbold{\chi}$ from the incompatibility equation and solving for
$\boldsymbol{f}$ from the equilibrium equation, for prescribed
$\boldsymbol{\alpha}$ on the given initial configuration. We refer to this
scheme as the elastic theory of continuously distributed dislocations (ECDD).
For the initial conditions on $\mathbold{\alpha}$ considered above, this step
is trivial, with ${\mathbold f}={\mathbold X}$, where ${\mathbold X}$ is the
position field on the initial configuration.

  * •

The model admits an arbitrary specification of $\dot{{\mathbold f}}$ at a
point to uniquely evolve ${\mathbold f}$ using (4f) in time, and this rate is
prescribed to vanish.

Interface conditions for plastic distortion rate: The interface jump condition
for the $\mathbold{\alpha}$ solve is $[[\mathbold{\alpha}\times{\mathbold
V}+\hat{{\mathbold L}}^{p}+l^{2}\hat{\gamma}\mathop{\rm
curl}\nolimits\mathbold{\alpha}]]\times{\mathbold n}=\bf 0$, and this gets
imposed in the finite element formulation. Here, $[[(\cdot)]]$ denotes the
jump in $(\cdot)$ across the interface. This interface condition allows for
the discontinuity of certain components of plastic strain rate across the
interface [6].

Traction continuity and natural interface conditions consistent with the PDE
for $\mathbold{\chi},\dot{{\mathbold f}}$ are also enforced by our finite
element formulation.

###  2.3 Constitutive relations

Constitutive relations in MFDM are required for the stress ${\mathbold T}$,
the plastic distortion rate ${\mathbold L}^{p}$, and the dislocation velocity
${\mathbold V}$. The details of the thermodynamically consistent constitutive
formulations can be found in Sec. 3.1 of [1].

Table 1 presents the constitutive relation for Cauchy stress and mesoscopic
core energy density for the material. Here, $\rho^{*}$ denotes the mass
density of the pure, un-stretched elastic lattice, and $\mathbb{C}$ is the
fourth-order elasticity tensor, while ${\mathbold E}^{e}$ is the elastic
Lagrangian strain tensor. $\Upsilon(\mathbold{\alpha})$ is a heuristic
representation of the averaged microscopic core energy at the mesoscale, and
$\epsilon$ is a constant that has physical dimensions of
$stress\times{length}^{2}$.

Saint-Venant-Kirchhoff Material | $\displaystyle\phi({\mathbold W})=\frac{1}{2\rho^{*}}{\mathbold E}^{e}:\mathbb{C}:{\mathbold E}^{e},\quad{\mathbold T}={\mathbold F}^{e}[\mathbb{C}:{\mathbold E}^{e}]{{\mathbold F}^{e}}^{T}$
---|---
Core energy density | $\displaystyle\Upsilon(\mathbold{\alpha})=\frac{1}{2\rho^{*}}\epsilon\mathbold{\alpha}:\mathbold{\alpha}$
Table 1: Constitutive relations for Cauchy stress and core energy density.

Table 2 shows the constitutive relations for plastic distortion rate.
Classical crystal plasticity is assumed here and the plastic distortion
${\mathbold L}^{p}$ is given by the sum of slipping distortions on prescribed
slip systems [1]. In Table 2, $(\cdot)_{sym}$ denotes the symmetric part of
$(\cdot)$, $\hat{\gamma}_{0}^{k}$ is a reference strain rate,
$\hat{\gamma}^{k}$ is the slipping rate due to statistically stored
dislocations on the $k^{th}$ slip system, $n_{sl}$ is the total number of slip
system, $sgn(\tau^{k})$ denotes the sign of the scalar $\tau^{k}$, $g$ is the
material strength, $m$ is the material rate sensitivity. ${\mathbold m}^{k}$
and ${\mathbold n}^{k}$ represents the slip direction and slip plane normal
for the $k^{th}$ slip system, in the current configuration. The resolved shear
stress on the $k^{th}$ slip system is given as $\tau^{k}={\mathbold
m}^{k}\cdot{\mathbold T}{\mathbold n}^{k}$. The slip direction and slip plane
normal in the current configuration are related to the unstretched slip
direction (${\mathbold m}^{k}_{0}$) and the slip plane normal (${\mathbold
n}^{k}_{0}$) by ${\mathbold m}^{k}={\mathbold F}^{e}{\mathbold m}^{k}_{0}$,
and ${\mathbold n}^{k}={{\mathbold F}^{e}}^{-T}{\mathbold n}^{k}_{0}$. The
classical plastic distortion rate $\hat{{\mathbold L}}^{p}$ from crystal
plasticity, is augmented with an additive term in $\mathop{\rm
curl}\nolimits\mathbold{\alpha}$, motivated from the mechanical dissipation
expression in [1, Eq. (8)], with a mobility coefficient given by $l^{2}$
multiplied by the average slipping rate of the total slip systems [1]. Here,
$l$ is a material length related to the gross modeling of mesoscale effects of
dislocation core energy and is defined as $l^{2}=\epsilon/g_{0}$, with $g_{0}$
being the initial strength (initial yield stress in shear).

Crystal plasticity | $\displaystyle\hat{{\mathbold L}}^{p}={\mathbold W}\left(\sum_{k}^{n_{sl}}\hat{\gamma}^{k}{\mathbold m}^{k}\otimes{\mathbold n}^{k}\right)_{\textit{sym}};\quad\hat{\gamma}^{k}=\textit{sgn}(\tau^{k})\hat{\gamma}_{0}^{k}\left(\frac{|\tau^{k}|}{g}\right)^{\frac{1}{m}}$
---|---
$\displaystyle\quad{\mathbold L}^{p}=\hat{{\mathbold
L}}^{p}+\frac{l^{2}}{n_{sl}}\sum_{k}^{n_{sl}}|\hat{\gamma}^{k}|\mathop{\rm
curl}\nolimits\mathbold{\alpha}$
Table 2: Constitutive relations for plastic strain rate ${\mathbold L}^{p}$.

Table 3 shows the constitutive relations for dislocation velocity. Here,
${\mathbold d}$ denotes the direction of dislocation velocity, and its
magnitude is given by $\zeta$, ${\mathbold T}^{\prime}$ denotes the deviatoric
stress tensor, ${\mathbold X}$ is the third-order alternating unit tensor,
$\mu$ is the shear modulus, $g$ is the material strength, $\eta$ is a non-
dimensional material constant in the empirical Taylor relationship for
macroscopic strength vs dislocation density, and $b$ is the Burgers vector
magnitude of a full dislocation in the crystalline material. For motivation
behind the constitutive statement for ${\mathbold V}$, the interested reader
is referred to [9, 5].

$\displaystyle{\mathbold T}^{{}^{\prime}}={\mathbold T}-\frac{tr({\mathbold T})}{3}{\mathbold I}$; | $\displaystyle{\mathbold a}=\frac{1}{3}{\mathbold X}(tr({\mathbold T}){\mathbold F}^{e}\mathbold{\alpha});$ | $\displaystyle{\mathbold c}={\mathbold X}({\mathbold T}^{{}^{\prime}}{\mathbold F}^{e}\mathbold{\alpha})$
---|---|---
$\displaystyle{\mathbold d}={\mathbold c}-\left({\mathbold c}-\frac{{\mathbold a}}{|{\mathbold a}|}\right)\frac{{\mathbold a}}{|{\mathbold a}|};$ | $\displaystyle{\mathbold V}=\zeta\frac{{\mathbold d}}{|{\mathbold d}|};$ | $\displaystyle\zeta=\frac{\mu^{2}\eta^{2}b}{g^{2}n_{sl}}\>\sum_{k}^{n_{sl}}|\hat{\gamma}^{k}|$
Table 3: Constitutive relations for dislocation velocity ${\mathbold V}$.

Table 4 shows the evolution equation for material strength ($g$). Here,
$g_{s}$ denotes the saturation strength, $h$ is the total hardening rate,
$\Theta_{0}$ is the Stage II hardening rate, $k_{0}$ (non-dimensional)
characterizes the hardening rate due to geometrically necessary dislocations
(GNDs) [4].

$\displaystyle\dot{g}=h(\boldsymbol{\alpha},g)\left(|\boldsymbol{F}^{e}\boldsymbol{\alpha}\times\boldsymbol{V}|+\sum_{k}^{n_{sl}}|\hat{\gamma}^{k}|\right);$ | $\displaystyle h(\boldsymbol{\alpha},g)=\frac{\mu^{2}\eta^{2}b}{2(g-g_{0})}k_{0}|\boldsymbol{\alpha}|+\Theta_{0}\left(\frac{g_{s}-g}{g_{s}-g_{0}}\right)$
---|---
Table 4: Constitutive relations for material strength $g$.

All parameters in our model, except $k_{0}$ and $l$, are part of the
constitutive structure of well-accepted models of classical crystal plasticity
theory. The parameter $k_{0}$ was introduced in [4]. The length scale $l$
simply controls the refinement of the GND microstructure and does not play a
physically significant role in our results.

##  3 Review of results from MFDM

The MFDM framework is based on the space-time averaging of microscopic
dislocation mechanics, and the obtained governing PDEs shown in (4) are closed
by the phenomenological specification of ${\mathbold L}^{p}$, ${\mathbold V}$,
${\mathbold T}$ and $g$. This approach has been quite successful in addressing
some challenging problems in modern plasticity theory related to the
computation of patterning [1], dislocation internal stress [11], size effects
in micropillar confined thin metal films [3], polygonization [2], and slip
transmission at grain boundaries [35, 36] among others [9, 6, 37, 28, 19, 7,
16, 20, 41, 40, 38, 42, 18, 45, 17, 14, 23], including long-standing and
recent fundamental challenges in the prediction of large-deformation,
dislocation mediated elastic and elastic-plastic response [1, 11, 2, 3]. The
benefits of using such a framework are:

  * •

It is difficult to deduce a physical connection between the plastic
strain/distortion in classical plasticity theory to the mechanics of
dislocation, beyond modeling in 1-D. The MFDM framework brings out an explicit
connection between the plastic strain rate and the motion and geometry of an
evolving microscopic array of dislocations. Obviously, this has many benefits,
even for a phenomenological specification of the macroscopic plastic strain
rate. Moreover, the MFDM framework has allowed for a first unification between
phenomenological $\textit{J}_{2}$ and crystal plasticity theories and
quantitative dislocation mechanics.

  * •

With a single extra material fitting parameter beyond a classical plasticity
model (and two in the finite deformation setting), the MFDM framework has
enabled a significant variety of phenomena to be modeled, in qualitative and
quantitative accord with experimental results.

##  4 Results and Discussion

The initial configuration of NMLs, as shown in Fig. 2 (a), is such that there
are 12 alternating layers of Cu and Nb. Each layer has a thickness of
$l=80\>nm$ with the width of the composite being $W=960\>nm$. The aspect ratio
of the NML composite is 1:2, with an undeformed height of the sample being
$L=1920\>nm$. The nominal compression loading rate is
$|\hat{e}|=0.001\,s^{-1}$, and at any time $t$, the nominal compressive strain
is $|e|=|\hat{e}|t$. The corresponding true compressive strain is denoted as
$|e_{T}|$. The average true compressive stress is denoted by $\sigma$, while
the average nominal compressive stress is denoted by $\sigma_{N}$. Plastically
unconstrained boundary conditions are applied to all four (external)
boundaries of the domain, for all results shown in this work. Quasi-static
force equilibrium is solved for all simulations reported here.

The boundary conditions for the material velocity for the simulations of
compression of NMLs are as follows:

  * •

$v_{2}=0$ at the bottom boundary of the domain.

  * •

$v_{2}=-|\hat{e}|L$ at the top boundary of the domain, where $L$ is the length
of the laminate in the (undeformed) initial configuration at $t=0$.

  * •

The applied traction in the horizontal direction is zero on both the top and
the bottom boundary of the domain.

  * •

$v_{1}=0$ at a single node of the bottom boundary. This along with the above
conditions suffices to constrain rigid motion.

As discussed earlier, we assume 3 slip systems for both Cu and Nb. One of the
reference slip directions is aligned along the layer parallel direction, while
the other two are defined to initially accommodate symmetric double slip with
respect to the loading axis. With reference to Fig. 2 (a), $\theta_{1}$ and
$\theta_{2}$ are both assumed to be $30^{\circ}$ for Cu, while for Nb, both
are $45^{\circ}$, unless mentioned otherwise.

Table 5 shows the material parameters used for the numerical simulations of
NMLs under compression in this study. The $g_{0}$ values for Cu and Nb are
taken from [46], and the saturation strength is assumed to be $g_{s}=9g_{0}$.
We use these material properties for all of our simulations unless otherwise
stated, although these values of initial and saturation yield strength for
single crystal materials seem somewhat out of the norm. Accordingly, we also
perform simulations with more commonly used values without a qualitative
change in results.

Parameter | $\hat{\gamma}_{0}$ | $m$ | $\eta$ | $b$ | $g_{0}$ | $g_{s}$ | $\Theta_{0}$ | $k_{0}$ | $l$ | $E$ | $\nu$
---|---|---|---|---|---|---|---|---|---|---|---
($s^{-1}$) |  |  | (Å) | (MPa) | (MPa) | (MPa) |  | ($\mu m$) | (GPa) |
Cu | 0.001 | 0.03 | $\frac{1}{3}$ | 2.556 | 210 | 1890 | 273 | 20 | $\sqrt{3}\times 0.1$ | 144.58 | 0.324
Nb | 0.001 | 0.03 | $\frac{1}{3}$ | 2.86 | 262.5 | 2362.5 | 198 | 20 | $\sqrt{3}\times 0.1$ | 110.25 | 0.392
Table 5: Material parameters for Cu and Nb.

###  4.1 Kink band formation

(a)  (b)

(c)

Figure 2: (a) Schematic of NML under compression with Cu layers shown in grey
color and Nb layers in black color, (b) stress-strain curves (both true and
engineering) from our simulation, (c) Experimental stress-strain curve (Figure
in (c) reprinted from [47] with permission from Elsevier).

Fig. 2 (b,c) shows the true and engineering stress-strain curve obtained from
our simulation and the experimentally obtained stress-strain curve from [47],
and it is clear that the numerically obtained stress-strain curves do not
match with the experimental ones. The yield strength obtained from our
simulations is between 0.45 - 0.5 GPa, while in the experimental results from
[47], it is close to 1.0 GPa. The strain-hardening response is more prominent
in our numerical result as compared to the experimental observation, where
after yielding it stays at a more-or-less constant level of stress. However,
in the current work or our past works like [2, 3], we have not employed any
fitting of material parameters to obtain the numerical results that are in
qualitative or quantitative agreement with experimental observations. This is
discussed in a broader context in Section 4.6, where we compare strain
gradient plasticity (SGP) based models and MFDM for modeling problems in
mesoscale plasticity.

Fig. 3 shows the field plot of geometrically necessary dislocation (GND)
density ($\rho_{g}=|\mathbold{\alpha}|/b\>(m^{-2})$) and the norm of
logarithmic strain tensor $|ln{\mathbold V}|$. Here, ${\mathbold V}$ is
obtained from the polar decomposition of the deformation gradient tensor
(${\mathbold F}={\mathbold V}{\mathbold R}$), and ${\mathbold F}$ is defined
with respect to the initial configuration at $t=0$. GNDs get generated in the
domain due to the inhomogeneity in the material response after yielding from
the material inhomogeneity of the laminate structure in NMLs, and the
imposition of jump condition on plastic strain rate at the interfaces.

(a)

(b)

(c)

(d)

Figure 3: (a) Field plot of GND density
$\rho_{g}=|\mathbold{\alpha}|/b\>(m^{-2})$ (b) $|ln{\mathbold V}|$ field plot,
both at $|e_{T}|=23\%$ and $|e_{N}|=25.9\%$. The dashed black lines in both
Figures show the deformed lines of the interface between Cu and Nb layers in
the current configuration. Figure 4: Comparison of true stress-strain curve
for different $g_{0}$, and with $g_{s}=9g_{0}$ in both cases.

It is evident from Fig. 3 that kink-band forms in accordance with the
experimental observations in [32, 47]. It is evident that the deformation
field localizes within the band, and the kink band is inclined (not
perpendicular) to the layering in NML as shown in Fig. 1. Also, as per Fig. 3
(b) that the principal stretches of ${\mathbold V}$ are maximum within the
kink band formed, due to intense deformation in the band region.

The inclination of the kink band shown in Fig. 3 with respect to the layer
normal (in the initial configuration) i.e. $\beta$ (shown in Fig. 1 (a)) is
estimated roughly to be $32^{\circ}$, and curiously it is close to
$\beta=35^{\circ}$ reported in the experimental work in [47]. The angle
$\beta$ of the inclination of the kink band is calculated based on the actual
deformed coordinates of the kink band boundary (note that the aspect ratio of
the plot box shown in Fig. 3 is close to $1:1$).

Fig. 4 shows the comparison of the stress-strain response with different
values of $g_{0}$ (initial yield), but with the same ratio between the
saturation yield strength and initial yield i.e. $g_{s}=9g_{0}$. As expected,
the initial yield strength for the NML is less, with reduced values of
$g_{0}$, as compared to the experimentally observed yield limit [47], but the
ultimate strength, in this case, is closer to the experimental value.
Importantly, we observe the formation of kink bands with similar shapes in our
numerical simulations for smaller values of $g_{0}$ and $g_{s}$, as well.

(a)

(b)

(c)

(d)

Figure 5: Rotation field of slip vectors ($\Delta\theta$ in degrees) in the
current configuration w.r.t. their initial orientation at $|e_{T}|=23\%$ for
(a) layer parallel slip vector, (b) $\theta_{1}$ oriented slip vector, (c)
$\theta_{2}$ oriented slip vector. The dashed black lines in all Figures show
the deformed lines of the interface between Cu and Nb layers in the current
configuration.

The orientation of the kink band depends on the Schmid factors of the 3 slip
systems considered, which is the largest for the 2 oblique slip systems. Based
on the deformed shape of the NML as shown in Fig. 5, the slip direction that
is inclined $(\theta_{1})$ w.r.t. to the layer parallel slip direction becomes
the most active slip system based on the Schmid factors in the deformed
lattice. However, it is not so clear what breaks the initial symmetry between
the double-symmetric slip systems to make the average resolved shear stress on
the $\theta_{1}$ oriented slip plane larger than the $\theta_{2}$ oriented
plane. The absolute value of the averaged Schmid factor (over the whole
domain) for the 3 slip systems in the initial configuration is 0 (layer-
parallel slip vector) and 0.4665 ($\theta_{1}$ and $\theta_{2}$ oriented slip
vectors), while the absolute values of the averaged Schmid factors over the
domain at $|e_{T}|=3.21\%$ are $0.0021$ (layer parallel slip vector), $0.4739$
($\theta_{1}$ oriented slip vector), and $0.4731$ ($\theta_{2}$ oriented slip
vector). At $|e_{T}|=\>23\%$, the absolute values of the averaged Schmid
factors for the 3 slip systems are $0.1407$ (layer-parallel slip vector),
$0.42246$ ($\theta_{1}$ oriented slip vector), and $0.3357$ ($\theta_{2}$
oriented slip vector).

The rotation of the deformed slip vectors w.r.t. their respective initial
orientations is shown in Fig. 5 at $|e_{T}|=23\%$, and it can be deduced that
near the kink band region, the rotation of slip vectors is relatively large.
The average rotation for the 3 slip vectors is $9.99^{\circ}$ (layer parallel
slip vector), $10.27^{\circ}$ ($\theta_{1}$ oriented slip vector), and
$9.64^{\circ}$ ($\theta_{2}$ oriented slip vector). The negative curl of the
inverse elastic distortion tensor is the GND density, and the rotation of
lattice vectors is a direct function of the (inverse) elastic distortion
tensor. As can be observed from Fig. 5, there are large gradients in the
rotation field of slip vectors near the kinked region, which results in
relatively large magnitudes of GND density in the kink band (refer to Fig. 3
(a)).

###  4.2 Effect of different slip systems, ordering of metallic layers, and
slip systems orientations

We report various numerical studies with respect to changing the ordering of
metallic layers, the effect of removal of slip vectors, and the slip vector
orientations on the kink band formation.

  * •

Flipping the order of Cu and Nb layers in the nano-laminate structure resulted
in a change in the orientation of the band as shown in Fig. 6 (a), due to a
geometrical asymmetry in the initial configuration.

  * •

Removal of $\theta_{1}$ inclined slip system, and just considering layer
parallel slip system and $\theta_{2}$ inclined slip system, resulted in the
orientation of the band as shown in Fig. 6 (b).

  * •

Removal of the $\theta_{2}$ oriented slip plane did not change the orientation
of the kink band formed, and it is similar to as shown in Fig. 3.

  * •

Removal of the layer parallel slip system and just considering the double
symmetric slip system with the same orientations, as before, did not change
the orientation of the kink band formed, as shown in Fig. 8.

  * •

We consider different orientations of slip vectors ($\theta_{1}=\theta_{2}$)
from $20^{\circ}$ to $45^{\circ}$ for Cu, and keep the slip vector orientation
for Nb to be the same i.e. $\theta_{1}=\theta_{2}=\>45^{\circ}$. Kink bands
form for slip vector orientations from $20^{\circ}$ to $30^{\circ}$, but for
$35^{\circ}$ to $45^{\circ}$ kink bands do not form, and NMLs undergo more
uniform compression.

(a)

(b)

(c)

Figure 6:  Field plot of GND density
$\rho_{g}=|\mathbold{\alpha}|/b\>(m^{-2})$ (a) after flipping the order of Cu
and Nb layers at $|e_{T}|=23\%$, (b) after removal of $\theta_{1}$ oriented
slip plane, and at $|e_{T}|=20.5\%$, but the order of metallic layers is the
same as shown in Fig. 2 (a). The dashed black lines in both Figures show the
deformed lines of the interface between Cu and Nb layers in the current
configuration. Figure 7: True stress-strain curve with/without accounting for
layer-parallel slip system.

The stress-strain response with just 2 slip systems is relatively stronger as
compared to the 3-slip system case, as shown in Fig. 7. As concerns, the
$\sim$ flat experimental stress-strain response beyond yielding, the
availability of more slip systems in realistic modeling of the NML would lead
to more plastic straining, but it would be accompanied by more gain in
strength as well. So, in general, it is difficult to say with certainty that
the availability of more slip systems will result in less strain-hardening,
but that is what we observe in our limited simulations reported here.

We further compare the field plots of GND density ($\rho_{g}$) with/without
the layer parallel slip system as shown in Fig. 8 at $|e_{T}|=23\%$ strain. It
is evident that the shape of the kink band and the GND density are similar in
the two cases. This is a departure from the results reported in the
experimental work by Zecevic et al. [46], where it was reported that the
layer-parallel slip system is responsible for the evolution of the shape of
the kink band. We leave a more realistic investigation of the effects of layer
parallel slip systems for future work, where we will be modeling the 3-D
geometry of NML and considering all the slip systems in Cu (FCC) and Nb (BCC).

Another interesting observation is that the domain-averaged rotation, with
respect to the initial orientation, decreases with an increase in the initial
orientation angle of the slip vectors from the loading axis, in the range of
initial orientations where kink bands form. This shows that in order to
accommodate the same level of plastic deformation, slip vectors with smaller
angular deviation from the loading axis rotate more on average during plastic
deformation.

(a)

(b)

(c)

Figure 8: (a) $\rho_{g}$ field plot for 2-slip system only (b) $\rho_{g}$
field plot for all 3-slip systems, both at $|e_{T}|=23\%$. The dashed black
lines in both Figures show the deformed lines of the interface between Cu and
Nb layers in the current configuration.

###  4.3 Effect of layer thickness length scale on the formation of kink band

We investigate the effect of layer thickness on the formation of kink bands in
NMLs. It was experimentally observed in the work by Nizolek et al. [32] that
kink bands do not form and homogeneous deformation is observed for layer
thickness greater than $250\>nm$. We run our simulation with the same
parameters as stated in Table 5, and take the layer thickness $l=500nm$, and
the NML is composed of a total of 12 layers. Hence, the width of the NML
composite is $W=6\>\mu m$, and keeping the same aspect ratio of 1:2, the
length of the NML composite is $L=12\>\mu m$. It can be deduced from the
deformed shape of NML (shown in Fig. 9 (b)) that kink bands do not form with
larger layer thickness, and it is getting compressed homogeneously without the
formation of any kink bands.

(a)

(b)

(c)

Figure 9: (a) True stress-strain curve comparison for layer thickness
$h=80\>nm$ and $h=500\>nm$, (b) $\rho_{g}$ field plot at $|e_{T}|=13.63\%$ for
layer thickness $h=500\>nm$. The dashed black lines in Figure (b) show the
deformed lines of the interface between Cu and Nb layers in the current
configuration.

###  4.4 Conventional plasticity within our model

In this section, we investigate the response of NML in three cases. For case
(i), we take the usual parameters ($k_{0}=20,l=0.01732,{\mathbold V}\neq\bf
0$), while for case (ii) the parameters are ($k_{0}=0,l=0,{\mathbold V}\neq\bf
0$), and for case (iii) the parameters taken are ($k_{0}=0,l=0,{\mathbold
V}=\bf 0$). Physically, case (ii) corresponds to just Voce’s law-based
hardening (without the effects of GND hardening) and no plastic strain rate
due to mesoscopic core energy. For case (iii), in addition to case (ii), there
is no plastic straining due to the motion of GNDs, and this case is closest to
conventional plasticity within our unified crystal plasticity and dislocation
mechanics framework. As expected, with no hardening from GNDs accounted for in
case (ii) and case (iii), we obtain a weaker strain-hardening response, as
compared to case (i). It was observed in numerical simulations that kink bands
do not form for case (ii) and case (iii), as shown in Fig. 10 (b), which shows
the GNDs density field plot for case (ii) at $|e_{T}|=12\%$ true strain.

(a)

(b)

(c)

Figure 10: (a) True stress-strain curve comparison for the 3 cases considered,
(b) $\rho_{g}$ field plot at $|e_{T}|=11.3\%$ for case (ii):
$k_{0}=0,\>l=0,\>{\mathbold V}\neq\bf 0$. The dashed black lines in Figure (b)
show the deformed lines of the interface between Cu and Nb layers in the
current configuration.

###  4.5 NML subjected to layer-perpendicular compression

Figure 11: True stress-strain curve when the layer direction is aligned
perpendicular or parallel to the compression loading direction.

Figure 12: $\rho_{g}$ field plot at $|e_{T}|=18.23\%$ strain, when the layer
direction is aligned perpendicular to the compression loading direction. The
dashed black lines in the Figure show the deformed lines of the interface
between Cu and Nb layers in the current configuration.

In this section, the layer direction of NML is aligned perpendicular to the
compression loading direction. As observed in the experiments [13], kink bands
do not form for layer perpendicular compression and the deformation of the NML
is shear-driven parallel to the layer direction. This results in material flow
under shearing within the individual layers as shown in Fig. 12. For the layer
perpendicular micropillar compression of NMLs in [13], it was reported that
diagonally inclined shear bands are formed, from the left or right boundary to
the right/left boundary. We leave such studies for future work where we model
the slender micropillar shape of the NML using a plane strain idealization
with the undulating interfaces. Up to the rotation of the layers towards the
loading axis, similar behavior is observed in the work [3].

###  4.6 Comparison of strain gradient plasticity (SGP) models and MFDM

  * •

The numerical results that we have obtained are with continuous plastic flow
across the interfaces, imposing continuity of certain components of plastic
strain rate across the interface. Moreover, we see the accumulation of GND
density across the interface (refer to Fig. 3 (a)), without having to do
constrained plastic flow across the interface, which models the impenetrable
interfaces in MFDM framework, as discussed in [36]. This is unlike the recent
work by Zecevic et al. [46], where ‘micro-hard’ boundary conditions are
implemented on the interfaces for their numerical study. From the current
state-of-the-art of SGP models [27, 46] there appears to be a high degree of
indeterminacy in the nature of boundary/interface conditions to be imposed,
with a significant impact on model predictions.

Using ‘micro-hard’ boundary conditions within SGP models would make it
difficult to reproduce drastically different scalings in the micropillar
compression experiments for two different configurations ($90^{\circ}$ and
$45^{\circ}$ oriented metal thin film with respect to compression loading
axis), as experimentally observed in [31]. In the work of Kuroda et al. [27],
the plastic constraints on the metal-ceramic interface are relaxed beyond a
certain level of plastic strain gradient on the interface/boundary, in order
to obtain the same scalings as experimentally observed. However, our model is
able to reproduce similar scalings as compared to experimental ones, without
any ad-hoc modifications to the boundary conditions, as shown in [3].
Moreover, our model also predicts the formation of kink bands in the
compression of NMLs, as shown in this current (first) simplified study, again
with no special fitting of material parameters or changes to the structure of
the theory (which, of course, includes the nature of boundary and interface
conditions). The jump conditions of our model have also been shown to be
successful in studies of texture evolution and recrystallization, among others
[28, 22, 39].

  * •

Our model is not able to reproduce size dependence on the initial yield
strength, as observed in the micropillar compression experiments of single
crystal pure Ni [43]; to our knowledge, SGP-based theories have not modeled
these experiments either. Most SGP theories are able to predict a significant
size effect at initial yield in the presence of boundary constraints or in the
presence of inhomogeneous deformation, without being able to fundamentally
distinguish boundary constraints arising from the kinematics of dislocation
slip [2, 3].

  * •

SGP studies of micropillar confined thin film plasticity and kink banding [26,
27, 46], usually employ a $\sim 0$ value of work hardening to reproduce
experimental behavior, raising the question of accurate representation of
macroscopic behavior by the models. Moreover, the SGP studies of [46, 34]
employ several variants of the core energy function and show a sensitive and
significant dependence on stress-strain response to such a choice. In our
work, no such choices are required, and the work hardening rates we use in our
strength evolution ensure, at least in the confined thin film problem and for
modeling macroscopic response, that a physically appropriate mechanical
response is obtained. For the present kink-band problem, the obtained stress-
strain response (without any fitting) is unsatisfactory compared to the
experiment (but no worse than the SGP result of [46]), and this is an issue
that requires further work.

##  5 Conclusion

This work reports a preliminary study of the modeling of kink-banding in NML
within the MFDM framework. We report reasonable success, which further
strengthens the ability of the MFDM framework to model important and
challenging problems in the field of modern plasticity. Our simulations of a
minimalistic model recover important experimental results in the compression
of NMLs such as:

  * •

dependence of kink-band formation on the layer thickness.

  * •

kink bands do not form when the layer direction is aligned perpendicular to
the compression loading direction, and there is shear-driven deformation
within the layers of NML.

At this time we are unable to provide a detailed analysis of the location of
the kink bands in our numerical simulations. Rough speculation in this regard
is as follows: the bands cannot cross the top and bottom boundaries because of
the applied velocity boundary conditions. The corners of the sample result in
stress concentrations due to geometric and changes in velocity boundary
conditions on intersecting external boundaries. These stress concentrations
may be the source of the location of the bands with the orientation governed
by the most active slip systems. Of course, there are subtle details of the
effects of the GND evolution that definitely affect the localization of the
band and the layer length scale effects. Such a detailed understanding of the
localization of the kink band is a fundamental challenge for future work as is
the modeling of the full 3-D geometry of NML samples and the crystallography
of actual slip systems present in Cu and Nb.

The prediction of experimentally observed kink bands in NMLs [32, 47] along
with the micropillar confined thin film plasticity [30, 31] form a discerning,
critical set of key experimental tests of mesoscale plasticity theories, to be
accomplished without ad-hoc model modifications on a case-by-case basis. The
MFDM model has been shown to be in qualitative accord with these experimental
tests, building confidence in its adequacy as a model for the intricate
phenomena involved in mesoscale plasticity.

## Acknowledgments

This work was supported by the grant NSF OIA-DMR $\\#2021019$.

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