Existence and uniqueness of bounded stable solutions to the Peierls–Nabarro model for curved dislocations

Abstract

We study the well-posedness of the vector-field Peierls–Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls–Nabarro model is reduced to a nonlocal scalar Ginzburg–Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson’s ratio $\nu \in (-{\frac{1}{2}}, {\frac{1}{3}})$. We then prove that any bounded stable solution to this nonlocal scalar Ginzburg–Landau equation has a 1D profile, which corresponds to the PDE version of flatness result for minimal surfaces with anisotropic nonlocal perimeter. Based on this, we finally obtain that steady states to the nonlocal scalar equation, as well as the original Peierls–Nabarro model, can be characterized as a one-parameter family of straight dislocation solutions to a rescaled 1D Ginzburg–Landau equation with the half Laplacian.

Appendices

Appendix

A Derivation of Euler–Lagrange equation

Proof of Lemma 2.1

From Definition 1 of local minimizers, we calculate the variation of the energy in terms of a perturbation with compact support in an arbitrary ball ${B_R}$. For any ${\mathbf {v}}\in C^\infty (B_R\backslash \Gamma )$ such that ${\mathbf {v}}$ has compact support in ${B_R}$ and satisfies (2.5), we consider the perturbation $\delta {\mathbf {v}}$, where $\delta$ is a small real number. We denote $\varepsilon :=\varepsilon ({\mathbf {u}})$, $\sigma :=\sigma ({\mathbf {u}})$ and $\varepsilon _{1}:=\varepsilon (\mathbf{v})$, $\sigma _{1}:=\sigma ({\mathbf {v}})$. Then we have that

$$\begin{aligned} \begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{\delta } (E({\mathbf {u}}+\delta {\mathbf {v}})-E({\mathbf {u}}))\\&\quad = \int _{B_R\backslash \Gamma }\frac{1}{2}(\sigma _{1}:\varepsilon + \sigma :\varepsilon _{1})\,\mathrm {d}x +\int _{{B_R}\cap \Gamma } \partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \,\mathrm {d}\Gamma \\&\quad =\int _{{B_R}\backslash \Gamma }\sigma :\varepsilon _{1}\,\mathrm {d}x +\int _{{B_R}\cap \Gamma } \partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \,\mathrm {d}\Gamma \\&\quad =\int _{B_R\backslash \Gamma }\sigma :\nabla {\mathbf {v}}\,\mathrm {d}x +\int _{B_R\cap \Gamma } \partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \,\mathrm {d}\Gamma \\&\quad =-\int _{B_R\backslash \Gamma }\partial _j\sigma _{ij} v_i\,\mathrm {d}x +\int _{B_R\cap \Gamma }\sigma _{ij}^{+} n_j^{+} v_i^{+} \,\mathrm {d}\Gamma \\&\qquad + \int _{B_R\cap \Gamma }\sigma _{ij}^{-} n_j^{-} v_i^{-} \,\mathrm {d}\Gamma +\int _{B_R\cap \Gamma }\partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \,\mathrm {d}\Gamma \ge 0, \end{aligned} \end{aligned}$$

where we used the property that $\sigma$ and $\nabla \cdot \sigma$ are locally integrable in $\{x_{3}>0\}\cup \{x_{3}<0\}$ when carrying out the integration by parts, and the outer normal vector of the boundary $\Gamma$ is ${\mathbf {n}}^{+}$ (resp. ${\mathbf {n}}^{-}$) for the upper half-plane (resp. lower half-plane). Similarly, taking perturbation as $-{\mathbf {v}}$, we have

$$\begin{aligned} \begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{\delta } (E({\mathbf {u}}-\delta {\mathbf {v}})-E({\mathbf {u}}))\\&\quad =\int _{B_R\backslash \Gamma }\partial _j\sigma _{ij} v_i\,\mathrm {d}x -\int _{B_R\cap \Gamma }\sigma _{ij}^{+} n_j^{+} v_i^{+} \,\mathrm {d}\Gamma \\&\qquad - \int _{B_R\cap \Gamma }\sigma _{ij}^{-} n_j^{-} v_i^{-} \,\mathrm {d}\Gamma -\int _{B_R\cap \Gamma } \partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \ge 0. \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned}&-\int _{B_R\backslash \Gamma }\partial _j\sigma _{ij} v_i\,\mathrm {d}x +\int _{B_R\cap \Gamma }\sigma _{ij}^{+} n_j^{+} v_i^{+} \,\mathrm {d}\Gamma \\&\quad + \int _{B_R\cap \Gamma }\sigma _{ij}^{-} n_j^{-} v_i^{-} \,\mathrm {d}x \,\mathrm {d}z +\int _{B_R\cap \Gamma }\partial _{1}W(u_{1}^{+}, u_{2}^{+})v_{1}^{+}+ \partial _{2} W(u_{1}^{+}, u_{2}^{+}) v_{2}^{+} \,\mathrm {d}\Gamma = 0. \end{aligned}$$

Noticing that ${\mathbf {n}}^{+}=(0,0,-1)$ and ${\mathbf {n}}^{-}=(0,0,1)$, we have

$$\begin{aligned} \begin{aligned}&\int _{B_R\cap \Gamma }\sigma _{ij}^{+} n_j^{+} v_i^{+} \,\mathrm {d}\Gamma + \int _{B_R\cap \Gamma }\sigma _{ij}^{-} n_j^{-} v_i^{-} \,\mathrm {d}\Gamma \\&\quad = \int _{B_R\cap \Gamma }-\sigma _{33}^{+} v_{3}^{+} \,\mathrm {d}x \,\mathrm {d}z+ \int _{B_R\cap \Gamma }\sigma _{33}^{-} v_{3}^{-} \,\mathrm {d}\Gamma + \int _{B_R\cap \Gamma }-\sigma _{13}^{+} v_{1}^{+} \,\mathrm {d}\Gamma + \int _{B_R\cap \Gamma }\sigma _{13}^{-} v_{1}^{-} \,\mathrm {d}\Gamma \\&\qquad + \int _{B_R\cap \Gamma }-\sigma _{23}^{+} v_{2}^{+} \,\mathrm {d}\Gamma + \int _{B_R\cap \Gamma }\sigma _{23}^{-} v_{2}^{-} \,\mathrm {d}\Gamma . \end{aligned} \end{aligned}$$

Recall that $v_{1}^{+}=-v_{1}^{-}$, $v_{3}^{+}=v_{3}^{-}$ and $v_{2}^{+}=-v_{2}^{-}$. Hence due to the arbitrariness of R, we conclude that the minimizer ${\mathbf {u}}$ must satisfy

$$\begin{aligned} \begin{aligned}&\int _{\Gamma }\left[ \sigma _{13}^{+} + \sigma _{13}^{-} -\partial _{1}W (u_{1}^{+}, u_{2}^{+})\right] v_{1}^{+} \,\mathrm {d}\Gamma =0,\\&\quad \int _{\Gamma }\left[ \sigma _{23}^{+} + \sigma _{23}^{-} -\partial _{2}W (u_{1}^{+}, u_{2}^{+})\right] v_{2}^{+} \,\mathrm {d}\Gamma =0,\\&\quad \int _\Gamma \left( \sigma _{33}^{+}-\sigma _{33}^{-}\right) v_{3}^{+} \,\mathrm {d}\Gamma =0,\\&\quad \int _{{\mathbb {R}}^2\backslash \Gamma } (\nabla \cdot \sigma ) \cdot {\mathbf {v}}~ \,\mathrm {d}x\,\mathrm {d}y \,\mathrm {d}z =0 \end{aligned} \end{aligned}$$

for any ${\mathbf {v}}\in C^\infty (B_R\backslash \Gamma )$ and $\mathbf{v}$ has compact support in $B_R$, which leads to the Euler–Lagrange equation (2.6). Here we write the equation $\nabla \cdot \sigma =0$ in ${\mathbb {R}}^2 \backslash \Gamma$ as the first equation of (2.6) in terms of the displacement ${\mathbf {u}}$, using the constitutive relation. $\square$

B Dirichlet to Neumann map

Proof of Lemma 2.2

Step 1. We take the Fourier transform of the elastic equations in (2.6) with respect to $x_{1},x_{2}$ and denote the corresponding Fourier variables as $k_{1}, k_{2}$.

Due to (2.3), ${\mathbf {u}}$ is unbounded and we take the Fourier transform for ${\mathbf {u}}$ with respect to $x_{1}, x_{2}$ by regarding them as tempered distributions. For notation simplicity, denote the Fourier transforms to be $\hat{{\mathbf {u}}}$. Let $k=(k_{1}, k_{2})$ and $|k|=\sqrt{k^2_{1}+k^2_{2}}$. We have

$$\begin{aligned}&(1-2\nu ) \partial _{33} {\hat{u}}_{1} - [(2-2\nu )k_{1}^2 + (1-2\nu )k_{2}^2] {\hat{u}}_{1} + i k_{1} \partial _{3} {\hat{u}}_{3} - k_{1} k_{2} {\hat{u}}_{2} = 0, \end{aligned}$$

(B.1)

$$\begin{aligned}&(2-2\nu )\partial _{33}{\hat{u}}_{3} - (1-2\nu )|k|^2{\hat{u}}_{3} + i k_{1} \partial _{3} {\hat{u}}_{1} + ik_{2} \partial _{3} {\hat{u}}_{2} = 0, \end{aligned}$$

(B.2)

$$\begin{aligned}&(1-2\nu ) \partial _{33} {\hat{u}}_{2} - [(2-2\nu )k_{2}^2 + (1-2\nu )k_{1}^2] {\hat{u}}_{2} + i k_{2} \partial _{3} {\hat{u}}_{3} - k_{1} k_{2} {\hat{u}}_{1} = 0. \end{aligned}$$

(B.3)

We can first eliminate ${\hat{u}}_{2}$ using (B.1), then eliminate ${\hat{u}}_{3}$ and obtain the ODE for ${\hat{u}}_{1}$

$$\begin{aligned} \partial _{3}^4 {\hat{u}}_{1} - 2 |k|^2 \partial _{3}^2 {\hat{u}}_{1}+ |k|^4 {\hat{u}}_{1}=0. \end{aligned}$$

Next we use this ODE for ${\hat{u}}_{1}$ to simplify (B.1), (B.2), and (B.3) again and then eliminate ${\hat{u}}_{1}$ and ${\hat{u}}_{2}$ together. We obtain the ODE for ${\hat{u}}_{3}$

$$\begin{aligned} \partial _{3}^4 {\hat{u}}_{3} - 2|k|^2 \partial _{3}^2 {\hat{u}}_{3} +|k|^4 {\hat{u}}_{3} = 0. \end{aligned}$$

By the symmetry of ${\hat{u}}_{1}$ and ${\hat{u}}_{2}$, we have the same ODE for ${\hat{u}}_{2}$.

We look for solutions whose derivatives have decay properties, which exclude exponentially growing solutions as $|x_{3}|\rightarrow +\infty$. Denote

$$\begin{aligned} {\hat{u}}_{1}^{-} = (A^{-} + B^{-}|k| x_{3})e^{|k|x_{3}}, \quad x_{3}<0, \end{aligned}$$

where $A^{-}, B^{-}$ are constants to be determined. Similarly, denote

$$\begin{aligned} {\hat{u}}_{3}^{-} = (C^{-} + D^{-}|k| x_{3})e^{|k|x_{3}}, \quad {\hat{u}}_{2}^{-} = (E^{-} + F^{-} x_{3}|k|)e^{|k|x_{3}}, \quad x_{3}<0, \end{aligned}$$

where $C^{-}, D^{-}, E^{-}, F^{-}$ are constants to be determined. For $x_{3}>0$, we have another six constants $A^{+}, B^{+}, C^{+}, D^{+}, E^{+}, F^{+}$ to be determined and for $x_{3}>0$,

$$\begin{aligned} \begin{aligned} {\hat{u}}_{1}^{+}&= (A^{+} - B^{+}|k| x_{3})e^{-|k|x_{3}}, \\ {\hat{u}}_{3}^{+}&= (C^{+} - D^{+} |k|x_{3})e^{-|k|x_{3}}, \\ {\hat{u}}_{2}^{+}&= (E^{+} - F^{+} |k|x_{3})e^{-|k|x_{3}}. \end{aligned} \end{aligned}$$

Step 2. Given the Dirichlet values of $u_{1}$ and $u_{2}$, we express all the other constants by $A^\pm$ and $E^\pm$.

First, plugging ${\hat{u}}_{1}^{-}$, ${\hat{u}}_{2}^{-}$, and ${\hat{u}}_{3}^{-}$ into (B.1), we have

$$\begin{aligned} (2-4\nu )|k|^2B^{-} - k_{1}^2 A^{-} + i k_{1} (C^{-}|k|+ D^{-} |k|)- k_{1} k_{2} E^{-}=0 \end{aligned}$$

and

$$\begin{aligned} -k_{1}^{2} B^{-}+i k_{1} D^{-}|k|-k_{1} k_{2} F^{-}=0. \end{aligned}$$

Plugging ${\hat{u}}_{1}^{-}$, ${\hat{u}}_{2}^{-}$, and ${\hat{u}}_{3}^{-}$ into (B.2), we have

$$\begin{aligned} |k|^{2} C^{-}+(4-4\nu )|k|^{2} D^{-}+i k_{1}|k| A^{-}+i k_{1}|k| B^{-}+i k_{2}|k| E^{-}+i k_{2}|k| F^{-}=0 \end{aligned}$$

and

$$\begin{aligned} |k|^2 D^{-} + i k_{1} |k|B^{-} + i k_{2} |k|F^{-} =0. \end{aligned}$$

Plugging ${\hat{u}}_{1}^{-}$, ${\hat{u}}_{2}^{-}$, and ${\hat{u}}_{3}^{-}$ into (B.3), we have

$$\begin{aligned} (2-4\nu )|k|^2F^{-} - k_{2}^2 E^{-} + i k_{2} (C^{-}|k|+ D^{-} |k|)- k_{1} k_{2} A^{-}=0 \end{aligned}$$

and

$$\begin{aligned} -k_{2}^{2} F^{-}+i k_{2} D^{-}|k|-k_{1} k_{2} B^{-}=0. \end{aligned}$$

Simplifying these relations gives us

$$\begin{aligned} \begin{aligned}&{B^{-}=\frac{i k_{1}}{|k|} D^{-}} ,\quad {F^{-}=\frac{i k_{2}}{|k|} D^{-}}, \\&{-k_{1} A^{-}-k_{2} E^{-}+i|k| C^{-}=(4 \nu -3) i|k| D^{-}.} \end{aligned} \end{aligned}$$

Combining this with the boundary symmetry (2.2), we have

$$\begin{aligned} A^{+} = - A^{-}, \ \ B^{+} = -B^{-},\ \ C^{+}= C^{-},\ \ D^{+}= D^{-},\ \ E^{+} = -E^{-}, \ \ F^{+}= -F^{-}. \end{aligned}$$

Then by $\sigma _{33}^{+}= \sigma _{33}^{-}$ on $\Gamma$ in (2.6), we further obtain $C^{-}= (2\nu -1)D^{-}$. Therefore, all the other constants can be expressed in terms of $A^{-}$ and $E^{-}$. In particular, we conclude that $\sigma _{13}(x_{1},x_{2}, 0^{+})$ and $\sigma _{23}(x_{1},x_{2}, 0^{+})$ can be expressed as in (2.7). $\square$