# Interface-Dominated Plasticity and Kink Bands in Metallic Nanolaminates

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## Abstract

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## 1. Introduction

**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 [8] with permission from Elsevier.).

**A**on a vector

**b**is denoted by $\mathit{A}\mathit{b}$. The inner product of two vectors is denoted by $\mathit{a}\xb7\mathit{b}$, while the inner product of two second-order tensors is denoted by $\mathit{A}:\mathit{B}$. 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 $(\xb7)$ w.r.t. the ${x}_{i}$ coordinate of this coordinate system is denoted by ${(\xb7)}_{,i}$, and ${e}_{i}$ denotes the unit vector in the ${x}_{i}$ direction. The time derivative of a quantity $(\xb7)$ is denoted by $\dot{(\xb7)}$. Einstein’s summation convention is always implied unless mentioned otherwise. The symbols $\mathrm{grad}$, $\mathrm{div}$, and $\mathrm{curl}$ denote the gradient, divergence, and curl on the current configuration, respectively. For a second order tensor

**A**, vectors

**v**,

**a**, and

**c**, and a spatially constant vector field

**b**, the operations of $\mathrm{div}$, $\mathrm{curl}$, and the cross-product of a tensor $(\times )$ with a vector are defined as follows:

**X**.

## 2. Review of MFDM Theory

**v**denotes the material velocity, $\mathit{L}=\mathrm{grad}\mathit{v}$ is the velocity gradient,

**T**is the (symmetric) Cauchy stress tensor, and

**V**denotes the dislocation velocity field with respect to the material. Equation (3a) is the incompatibility equation for the inverse elastic distortion tensor, while Equation (3b) is derived from the statement of conservation of the Burgers vector content of an arbitrary area patch in the current configuration [18]. Equation (3c) is the linear momentum balance for both quasistatic and dynamic cases.

**W**is introduced, expressing it as a sum of a gradient $\left(\mathrm{grad}\mathit{f}\right)$ and a divergence-free part ($\mathit{\chi}$) [11]. This decomposition replaces the tensor field

**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 [15].

**f**denotes the plastic position vector, and the evolution equation for

**f**is given by (4c). ${\mathit{L}}^{p}$ is the additional mesoscale field which represents the averaged rate of plastic straining due to all dislocations that cannot be represented by $\mathit{\alpha}\times \mathit{V}$, 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 ${\mathit{L}}^{p}$, as there is no corresponding field in FDM theory to ${\mathit{L}}^{p}$.

**v**are specified, for a well-set evolution in MFDM.

#### 2.1. Boundary Conditions

- The $\mathit{\alpha}$ evolution equation (shown in Equation (4a)) has a convective boundary condition of the form $(\mathit{\alpha}\times \mathit{V}+{\mathit{L}}^{p})\times \mathit{n}=\mathit{\Phi}$, where $\mathit{\Phi}$ 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 $\mathit{\Phi}\mathit{n}=\mathbf{0}$. Here,
**n**is the outward unit normal field on the boundary.The boundary condition is specified in two ways: (a) Plastically constrained: $\mathit{\Phi}(x,t)=\mathbf{0}$ is specified at a point**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 ${\widehat{\mathit{L}}}^{p}\times \mathit{n}$ is simply evaluated at the boundary (akin to an outflow condition), along with the specification of dislocation flux $\mathit{\alpha}(\mathit{V}\xb7\mathit{n})$ on the inflow part of the boundary. Additionally, for all calculations presented in this paper $(\mathrm{curl}\mathit{\alpha}\times \mathit{n})=\mathbf{0}$ 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, $\mathit{\chi}\mathit{n}=\mathbf{0}$ is applied on the outer boundary of the domain, which along with the system (4b) ensures that $\mathit{\chi}$ vanishes when $\mathit{\alpha}$ is zero in the entire domain.
- The $\mathit{f}$ evolution equation requires a Neumann boundary condition, i.e., $\left(\mathrm{grad}\dot{\mathit{f}}\right)\mathit{n}=(\mathit{\alpha}\times \mathit{V}+{\mathit{L}}^{p}-\dot{\mathit{\chi}}-\mathit{\chi}\mathit{L})\mathit{n}$ 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 $\mathit{\alpha}(\mathit{x},0)=\mathbf{0}$ is assumed for all simulations here.
- In general, the initial condition for $\mathit{f}$ is obtained by solving for $\mathit{\chi}$ from the incompatibility equation and solving for $\mathit{f}$ from the equilibrium equation, for prescribed $\mathit{\alpha}$ on the given initial configuration. We refer to this scheme as the elastic theory of continuously distributed dislocations (ECDD). For the initial conditions on $\mathit{\alpha}$ considered above, this step is trivial, with $\mathit{f}=\mathit{X}$, where
**X**is the position field on the initial configuration. - The model admits an arbitrary specification of $\dot{\mathit{f}}$ at a point to uniquely evolve
**f**using (4c) in time, and this rate is prescribed to vanish.

#### 2.3. Constitutive Relations

**T**, the plastic distortion rate ${\mathit{L}}^{p}$, and the dislocation velocity

**V**. The details of the thermodynamically consistent constitutive formulations can be found in Section 3.1 of [11].

**d**denotes the direction of dislocation velocity, and its magnitude is given by $\zeta $, ${\mathit{T}}^{\prime}$ denotes the deviatoric stress tensor,

**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 the motivation behind the constitutive statement for

**V**, the interested reader is referred to [20,21].

## 3. Review of Results from MFDM

**V**,

**T**and g. This approach has been quite successful in addressing some challenging problems in modern plasticity theory related to the computation of patterning [11], dislocation internal stress [15], size effects in micropillar confined thin metal films [23], polygonization [16], and slip transmission at grain boundaries [24,25] among others [18,19,20,26,27,28,29,30,31,32,33,34,35,36,37,38,39], including long-standing and recent fundamental challenges in the prediction of large-deformation, dislocation mediated elastic and elastic-plastic response [11,15,16,23]. The benefits of using such a framework are as follows:

- 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 ${\mathit{J}}_{2}$ 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

- ${v}_{2}=0$ at the bottom boundary of the domain.
- ${v}_{2}=-\left|\widehat{e}\right|L$ 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.

#### 4.1. Kink Band Formation

**V**is obtained from the polar decomposition of the deformation gradient tensor ($\mathit{F}=\mathit{V}\mathit{R}$), and

**F**is defined with respect to the initial configuration at $t=0$. GNDs are 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.

**V**are maximum within the kink band formed, due to intense deformation in the band region.

#### 4.2. Effect of Different Slip Systems, Ordering of Metallic Layers, and Slip Systems Orientations

- 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 Figure 6a, 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 Figure 6b.
- 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 Figure 3.
- 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., ${\theta}_{1}={\theta}_{2}=\phantom{\rule{0.222222em}{0ex}}{45}^{\circ}$. 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.

#### 4.3. Effect of Layer Thickness Length Scale on the Formation of Kink Band

#### 4.4. Conventional Plasticity within Our Model

#### 4.5. NML Subjected to Layer-Perpendicular Compression

#### 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 Figure 3a), without having to do constrained plastic flow across the interface, which models the impenetrable interfaces in MFDM framework, as discussed in [25]. This is unlike the recent work by Zecevic et al. [5], where ‘micro-hard’ boundary conditions are implemented on the interfaces for their numerical study. From the current state-of-the-art of SGP models [5,40] 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 [41]. In the work of Kuroda et al. [40], 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 [23]. 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 [27,42,43].
- 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 [44]; 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 [16,23].
- SGP studies of micropillar confined thin film plasticity and kink banding [5,40,45], 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 [5,46] 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 [5]), and this is an issue that requires further work.

## 5. Conclusions

- 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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**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 [8] with permission from Elsevier).

**Figure 3.**(

**a**) Field plot of GND density ${\rho}_{g}=\left|\mathit{\alpha}\right|/b\phantom{\rule{0.222222em}{0ex}}\left({m}^{-2}\right)$ (

**b**) $\left|ln\mathit{V}\right|$ 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 the Cu and Nb layers in the current configuration.

**Figure 4.**Comparison of true stress-strain curve for different ${g}_{0}$, and with ${g}_{s}=9{g}_{0}$ in both cases.

**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 the Cu and Nb layers in the current configuration.

**Figure 6.**Field plot of GND density ${\rho}_{g}=\left|\mathit{\alpha}\right|/b\phantom{\rule{0.222222em}{0ex}}\left({m}^{-2}\right)$ (

**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 Figure 2a. The dashed black lines in both Figures show the deformed lines of the interface between the Cu and Nb layers in the current configuration.

**Figure 8.**(

**a**) ${\rho}_{g}$ field plot for 2 slip systems 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 the Cu and Nb layers in the current configuration.

**Figure 9.**(

**a**) True stress-strain curve comparison for layer thickness $h=80\phantom{\rule{0.222222em}{0ex}}$ nm and $h=500\phantom{\rule{0.222222em}{0ex}}$ nm, (

**b**) ${\rho}_{g}$ field plot at $|{e}_{T}|=13.63\%$ for layer thickness $h=500\phantom{\rule{0.222222em}{0ex}}$ nm. The dashed black lines in subfigure (

**b**) show the deformed lines of the interface between the Cu and Nb layers in the current configuration.

**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): ${k}_{0}=0,\phantom{\rule{0.222222em}{0ex}}l=0,\phantom{\rule{0.222222em}{0ex}}\mathit{V}\ne \mathbf{0}$. The dashed black lines in subfigure (

**b**) show the deformed lines of the interface between the Cu and Nb layers in the current configuration.

**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 the Cu and Nb layers in the current configuration.

Saint-Venant-Kirchhoff Material | $\varphi \left(\mathit{W}\right)=\frac{1}{2{\rho}^{*}}{\mathit{E}}^{\mathit{e}}:\mathbb{C}:{\mathit{E}}^{\mathit{e}},\phantom{\rule{1.em}{0ex}}\mathit{T}={\mathit{F}}^{\mathit{e}}[\mathbb{C}:{\mathit{E}}^{\mathit{e}}]{{\mathit{F}}^{\mathit{e}}}^{\mathit{T}}$ |

Core energy density | $\rm Y}\left(\mathit{\alpha}\right)=\frac{1}{2{\rho}^{*}}\u03f5\mathit{\alpha}:\mathit{\alpha$ |

Crystal plasticity | $\widehat{\mathit{L}}}^{\mathit{p}}=\mathit{W}{\left(\right)}_{\sum _{\mathit{k}}^{{\mathit{n}}_{\mathit{s}\mathit{l}}}}\mathit{sym};\phantom{\rule{1.em}{0ex}}{\widehat{\gamma}}^{\mathit{k}}=\mathit{sgn}\left({\tau}^{\mathit{k}}\right){\widehat{\gamma}}_{0}^{\mathit{k}}{\left(\right)}^{\frac{|{\tau}^{\mathit{k}}|}{\mathit{g}}}\frac{1}{\mathit{m}$ |

$\phantom{\rule{1.em}{0ex}}{\mathit{L}}^{p}={\widehat{\mathit{L}}}^{p}+\frac{{l}^{2}}{{n}_{sl}}\sum _{k}^{{n}_{sl}}\left|{\widehat{\gamma}}^{k}\right|\mathrm{curl}\mathit{\alpha}$ |

$\mathit{T}}^{{}^{\prime}}=\mathit{T}-\frac{\mathit{t}\mathit{r}\left(\mathit{T}\right)}{3}\mathit{I$; | $\mathit{a}=\frac{1}{3}\mathit{X}\left(\mathit{t}\mathit{r}\left(\mathit{T}\right){\mathit{F}}^{\mathit{e}}\mathit{\alpha}\right);$ | $\mathit{c}=\mathit{X}\left({\mathit{T}}^{{}^{\prime}}{\mathit{F}}^{\mathit{e}}\mathit{\alpha}\right)$ |

$\mathit{d}=\mathit{c}-\left(\right)open="("\; close=")">\mathit{c}-\frac{\mathit{a}}{\left|\mathit{a}\right|}\frac{\mathit{a}}{\left|\mathit{a}\right|};$ | $\mathit{V}=\zeta \frac{\mathit{d}}{\left|\mathit{d}\right|};$ | $\zeta =\frac{{\mu}^{2}{\eta}^{2}b}{{g}^{2}{n}_{sl}}\phantom{\rule{0.222222em}{0ex}}\sum _{k}^{{n}_{sl}}\left|{\widehat{\gamma}}^{k}\right|$ |

$\dot{\mathit{g}}=\mathit{h}(\mathit{\alpha},\mathit{g})\left(\right)open="("\; close=")">|{\mathit{F}}^{\mathit{e}}\mathit{\alpha}\times \mathit{V}|+\sum _{\mathit{k}}^{{\mathit{n}}_{\mathit{s}\mathit{l}}}\left|{\widehat{\gamma}}^{\mathit{k}}\right|;$ | $\mathit{h}(\mathit{\alpha},\mathit{g})=\frac{{\mathit{\mu}}^{2}{\mathit{\eta}}^{2}\mathit{b}}{2(\mathit{g}-{\mathit{g}}_{0})}{\mathit{k}}_{0}\left|\mathit{\alpha}\right|+{\mathit{\Theta}}_{0}\left(\right)open="("\; close=")">\frac{{\mathit{g}}_{\mathit{s}}-\mathit{g}}{{\mathit{g}}_{\mathit{s}}-{\mathit{g}}_{0}}$ |

Parameter | ${\widehat{\gamma}}_{0}$ | m | $\eta $ | b | ${\mathit{g}}_{0}$ | ${\mathit{g}}_{s}$ | ${\mathit{\Theta}}_{0}$ | ${\mathit{k}}_{0}$ | l | E | $\mathit{\nu}$ |

(${s}^{-1}$) | (Å) | (MPa) | (MPa) | (MPa) | ($\mathsf{\mu}$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 |

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## Share and Cite

**MDPI and ACS Style**

Arora, A.; Arora, R.; Acharya, A.
Interface-Dominated Plasticity and Kink Bands in Metallic Nanolaminates. *Crystals* **2023**, *13*, 828.
https://doi.org/10.3390/cryst13050828

**AMA Style**

Arora A, Arora R, Acharya A.
Interface-Dominated Plasticity and Kink Bands in Metallic Nanolaminates. *Crystals*. 2023; 13(5):828.
https://doi.org/10.3390/cryst13050828

**Chicago/Turabian Style**

Arora, Abhishek, Rajat Arora, and Amit Acharya.
2023. "Interface-Dominated Plasticity and Kink Bands in Metallic Nanolaminates" *Crystals* 13, no. 5: 828.
https://doi.org/10.3390/cryst13050828