Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple

Abstract

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

1. Introduction

Mansfield [1] first introduced the concept of neutrality in which carefully designed holes and inclusions could be introduced into an existing elastic field without disturbing the original stress distribution in the uncut field. Accordingly, such holes and inclusions are referred to as ‘neutral holes' or ‘neutral inclusions’. The design of neutral inclusions usually involves the introduction of a thin or thick coating between the embedded inclusion and the surrounding elastic phase commonly referred to as the ‘matrix' [2–9]. Using the idea of multi-coating, Ammari et al. [10,11] constructed near-cloaking structures which nullify the first generalized polarization tensors (GPTs) of the cloaking device in the case of the conductivity problem or the first scattering coefficients in the case of the scattering problem. The analogous ideas behind acoustic (or band gap) and thermal cloaking have also been intensively investigated [12–17].

It is well-known that the presence of dislocations or Eshelby inclusions undergoing uniform eigenstrains will in general induce stress disturbance everywhere in a laminated or fibrous composite [18–20]. In this paper, we first use the ideas behind neutral inclusions to design novel cloaking lamellar structures for a screw dislocation dipole. The lamellar structure consists of a middle coating (or a lamellar inclusion) with locally variable thickness sandwiched between the left and the right elastic half-planes. A screw dislocation dipole is applied in the middle coating. The novelty of the present cloaking structures lies in the fact that the coating containing the dislocation dipole extends infinitely in both upward and downward directions. The shapes of the two interfaces and the locations of the two components of the dislocation dipole are judiciously designed in such a way that the prescribed uniform anti-plane stress fields in both half-planes are not disturbed by the dislocation dipole in the coating. Our design for the cloaking structures is based heavily on the construction of a conformal mapping function which maps the coating with variable thickness in the physical plane onto a strip in the image plane. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed design method. Using a similar method, we then design cloaking lamellar structures in the case of a circular Eshelby inclusion undergoing either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains embedded in the middle coating. The presence of the Eshelby inclusion will not induce any stress disturbance in the surrounding two half-planes. Finally, as a consequence of our results, we are also able to design cloaking lamellar structures for a concentrated couple in the middle coating.

2. Complex variable formulation

We establish a Cartesian coordinate system $\{x_i\}\hspace{0.17em}(i=1,2,3)$. For the in-plane deformations of an isotropic elastic material, the three in-plane stresses $({\sigma }_{11},\hspace{0.17em}{\sigma }_{22},\hspace{0.17em}{\sigma }_{12})$, two in-plane displacements $(u_1,\hspace{0.17em}{u}_2)$ and two stress functions $({\varphi }_1,\hspace{0.17em}{\varphi }_2)$ are given in terms of two analytic functions φ(z) and ψ(z) of the complex variable z = x₁ + ix₂ as [21,22]

$$\begin{array}{rl}& {\sigma }_{11}+{\sigma }_{22}=2[{\phi }^{\mathrm{\prime }}(z)+\overline{{\phi }^{\mathrm{\prime }}(z)}],\\ & {\sigma }_{22}-{\sigma }_{11}+2\text{i}{\sigma }_{12}=2[\overline{z}{\phi }^{\mathrm{\prime }\mathrm{\prime }}(z)+{\psi }^{\mathrm{\prime }}(z)],\end{array}$$ 2.1

and

$$\begin{array}{rl}& 2\mu (u_1+\text{i}{u}_2)=\kappa \phi (z)-z\overline{{\phi }^{\mathrm{\prime }}(z)}-\overline{\psi (z)},\\ & {\varphi }_1+\text{i}{\varphi }_2=\text{i}[\phi (z)+z\overline{{\phi }^{\mathrm{\prime }}(z)}+\overline{\psi (z)}],\end{array}$$ 2.2

where κ = 3 − 4ν for plane strain, κ = (3 − ν)/(1 + ν) for plane stress and μ, ν(0 ≤ ν ≤ 1/2) are the shear modulus and Poisson's ratio, respectively. In addition, the in-plane stresses are related to the two stress functions through [22]

$$\begin{array}{rl}& {\sigma }_{11}=-{\varphi }_{1,2},{\sigma }_{12}={\varphi }_{1,1},\\ & {\sigma }_{21}=-{\varphi }_{2,2},{\sigma }_{22}={\varphi }_{2,1}.\end{array}$$ 2.3

In the anti-plane shear deformations of an isotropic elastic material, the two anti-plane shear stress components σ₃₁ and σ₃₂, the out-of-plane displacement u₃ and the stress function ϕ₃ can be expressed in terms of a single analytic function f(z) of the complex variable z = x₁ + ix₂ as [22]

$${\sigma }_{32}+\text{i}{\sigma }_{31}=\mu f^{\mathrm{\prime }}(z),{\varphi }_3+\text{i}\mu u_3=\mu f(z).$$ 2.4

In addition, the two anti-plane stress components can be expressed in terms of the single stress function as [22]

$${\sigma }_{32}={\varphi }_{3,1},{\sigma }_{31}=-{\varphi }_{3,2}.$$ 2.5

3. Cloaking lamellar structures for a screw dislocation dipole

We consider a three-phase lamellar structure as shown in figure 1. Let $S_1,\hspace{0.17em}{S}_2$ and S₃ denote the left half-plane, the middle coating and the right half-plane, respectively, all of which are perfectly bonded through the two locally curved interfaces L₁ and L₂. A screw dislocation dipole is present in the coating. The dipole component with Burgers vector b is located at z = z₁ while the other dipole component with Burgers vector −b is located at z = z₂. The two half-planes S₁ and S₃ have the same shear modulus μ₁, and the coating S₂ has the modulus μ₂. Throughout the paper, the subscripts 1, 2 and 3 are used to identify the respective quantities in S₁, S₂ and S₃. At this stage, both the shapes of the two interfaces L₁ and L₂ and the locations of the two components of the screw dislocation dipole are unknown. We will endeavour to properly design the shapes of the two interfaces as well as the locations of the two components of the screw dislocation dipole in such a way that the prescribed uniform anti-plane stress fields inside both S₁ and S₃ are undisturbed even in the presence of the screw dislocation dipole in the middle coating. In other words, the stress disturbance due to the dislocation dipole is strictly confined to the middle coating and will not extend to the two surrounding half-planes through the designed interfaces.

Figure 1.

The cloaking lamellar structure for a screw dislocation dipole in the middle coating. (Online version in colour.)

We introduce the following general form of the conformal mapping function for the middle coating

$$z=\omega (\xi ),\xi ={\omega }^{-1}(z),H_1\le \mathrm{Re}\{\xi \}\le H_2,H_2>H_1>0.$$ 3.1

As shown in figure 2, using the mapping function in equation (3.1), the coating S₂ is mapped onto the strip: $H_1\le \mathrm{Re}\{\xi \}\le H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the left interface L₁ is mapped onto the vertical line: $\mathrm{Re}\{\xi \}=H_1,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the right interface L₂ is mapped onto another vertical line: $\mathrm{Re}\{\xi \}=H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the location of the dipole component at z = z₁ is mapped onto the point ξ = ξ₁ and the location of the other dipole component at z = z₂ is mapped onto the point ξ = ξ₂. The specific form of the mapping function will be determined through the satisfaction of the continuity conditions across the two perfect interfaces L₁ and L₂.

Figure 2.

The ξ-plane. (Online version in colour.)

By considering the fact that the prescribed uniform anti-plane stress fields in both the left and right half-planes S₁ and S₃ are uniform, we have

$$\begin{array}{rl}{f}_1(z)& =Cz,z\in S_1;\\ f_3(z)& =Cz+C_0,z\in S_3,\end{array}$$ 3.2

where C is a given complex number, and C₀ is a complex number to be determined.

By enforcing the continuity conditions of traction and displacement across the left and right perfect interfaces L₁ and L₂ with the aid of equation (3.2), we arrive at the following condition

$$\begin{array}{rl}{f}_2(\xi )& =f_2(\omega (\xi ))=\frac{C(\mathit{\Gamma }+1)}{2}\omega (\xi )+\frac{\overline{C}(\mathit{\Gamma }-1)}{2}\overline{\omega }(2H_1-\xi )\\ & =\frac{C(\mathit{\Gamma }+1)}{2}\omega (\xi )+\frac{\overline{C}(\mathit{\Gamma }-1)}{2}\overline{\omega }(2H_2-\xi )+\frac{C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)}{2},H_1\le \mathrm{Re}\{\xi \}\le H_2,\end{array}$$ 3.3

where Γ = μ₁/μ₂. In writing equation (3.3), ω(ξ) has been extended to the two regions $2H_1-H_2\le \mathrm{Re}\{\xi \}\le H_1$ and $H_2\le \mathrm{Re}\{\xi \}\le 2H_2-H_1$ using a similar technique to that adopted by Milton and Serkov [6], Jarczyk and Mityushev [7], Ru [23] and Suo [24]. A detailed derivation of equation (3.3) can be found in the appendix.

It follows from the above that when Γ ≠ 1, the following relationship can be obtained

$$\overline{\omega }(2H_1-\xi )=\overline{\omega }(2H_2-\xi )+\frac{C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)}{\overline{C}(\mathit{\Gamma }-1)},H_1\le \mathrm{Re}\{\xi \}\le H_2.$$ 3.4

After inspecting the above equation and carefully considering the conditions dictated by the problem at hand, a process inspired by the image method adopted by Chou [18] eventually leads us to a mapping function judiciously constructed as follows:

$$\begin{array}{rl}z& =\omega (\xi )=\xi +p\sum _{n=1}^{+\mathrm{\infty }}\ln \frac{\xi +{\overline{\xi }}_1-2H_1+2(n-1)(H_2-H_1)}{\xi +{\overline{\xi }}_2-2H_1+2(n-1)(H_2-H_1)}\\ & +p\sum _{n=1}^{+\mathrm{\infty }}\ln \frac{\xi +{\overline{\xi }}_1-2H_2+2(n-1)(H_1-H_2)}{\xi +{\overline{\xi }}_2-2H_2+2(n-1)(H_1-H_2)},\\ \xi & ={\omega }^{-1}(z),\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2,\end{array}$$ 3.5

where ${\xi }_1={\omega }^{-1}(z_1),\hspace{0.17em}{\xi }_2={\omega }^{-1}(z_2)$ or conversely $z_1=\omega ({\xi }_1),\hspace{0.17em}{z}_2=\omega ({\xi }_2)$, and p is a complex constant. All of the logarithmic singularities in the above mapping function are located outside the strip: $H_1\le \mathrm{Re}\{\xi \}\le H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$. More specifically, the logarithmic singularities in the first series in equation (3.5) are located on the left-hand side of the vertical line: $\mathrm{Re}\{\xi \}=H_1,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$ while the logarithmic singularities in the second series in equation (3.5) are located on the right-hand side of the vertical line: $\mathrm{Re}\{\xi \}=H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$. A detailed derivation of equation (3.5) can also be found in the Appendix. It is seen from equation (3.5) that z = ω(ξ) ≅ ξ as |ξ| → ∞. Thus, as | z| → ∞, the coating extends vertically and has a constant thickness given by H₂ − H₁. In this study, we will treat H₁ having the dimension of length as a scaling constant. Consequently, the mapping function in equation (3.5) contains in total of four parameters having the dimension of length: $H_2,\hspace{0.17em}p,\hspace{0.17em}{\xi }_1,\hspace{0.17em}{\xi }_2$. Here, we stress that the mapping function in equation (3.5) is constructed on the condition that Γ ≠ 1. Using the present method, it would seem to be impossible to design a cloaking lamellar structure for the case in which the left and right half-planes have different shear moduli. It is also difficult to design a cloaking lamellar structure for a single screw dislocation applied in the coating.

Remark 1. Since n is taken sufficiently large, the behaviour of the terms appearing in each of the individual series appearing in equation (3.5) can be written as follows:

$$\begin{array}{rl}& \ln \frac{\xi +{\overline{\xi }}_1-2H_1+2(n-1)(H_2-H_1)}{\xi +{\overline{\xi }}_2-2H_1+2(n-1)(H_2-H_1)}\approx \frac{{\overline{\xi }}_1-{\overline{\xi }}_2}{2(n-1)(H_2-H_1)}-\frac{{({\overline{\xi }}_2-{\overline{\xi }}_1)}^2}{8{(n-1)}^2{(H_2-H_1)}^2},\\ & \ln \frac{\xi +{\overline{\xi }}_1-2H_2+2(n-1)(H_1-H_2)}{\xi +{\overline{\xi }}_2-2H_2+2(n-1)(H_1-H_2)}\approx -\frac{{\overline{\xi }}_1-{\overline{\xi }}_2}{2(n-1)(H_2-H_1)}-\frac{{({\overline{\xi }}_2-{\overline{\xi }}_1)}^2}{8{(n-1)}^2{(H_2-H_1)}^2}.\end{array}$$

Consequently, the divergent part of each of the series in equation (3.5) ‘cancels' when summing the entire series in equation (3.5) resulting in a convergent series representing the mapping function.

In addition, we have from equations (3.4) and (3.5) that

$$C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)=2\overline{C}(\mathit{\Gamma }-1)\hspace{0.17em}(H_1-H_2),$$ 3.6

from which the complex constant C₀ can be uniquely determined as

$$C_0=(\mathit{\Gamma }-1)(H_1-H_2)\left(\frac{\mathrm{Re}\{C\}}{\mathit{\Gamma }}-\text{i}\text{Im}\{C\}\right).$$ 3.7

It is further deduced from equations (3.4) and (3.5) that

$$p=\frac{b}{\pi C(\mathit{\Gamma }-1)},$$ 3.8

which implies that the complex nature of p comes solely from the complex nature of C. The condition that Γ ≠ 1 can also be observed from equation (3.8). In addition, p is real when C is real, and p is purely imaginary when C is purely imaginary.

It is seen from equations (3.3) and (3.5) that

$$f_2(z)\cong \frac{C(\mathit{\Gamma }+1)-\overline{C}(\mathit{\Gamma }-1)}{2}z+O(1),|z|\to \mathrm{\infty }.$$ 3.9

The above fact implies that if the prescribed uniform stresses in S₁ and S₃ are ${\sigma }_{31}={\sigma }_{31}^0,\hspace{0.17em}{\sigma }_{32}={\sigma }_{32}^0$, the stress state at infinity in the coating is

$${\sigma }_{31}\cong {\sigma }_{31}^0,\hspace{0.17em}{\sigma }_{32}\cong \frac{{\sigma }_{32}^0}{\mathit{\Gamma }}\text{as}\hspace{0.17em}|z|\to \mathrm{\infty },$$ 3.10

which implies that the continuity conditions of traction and displacement across the two interfaces L₁ and L₂ have indeed been satisfied at infinity. In addition, it is seen from equation (3.10) that the coating is subjected to uniform remote stress ${\sigma }_{32}^{\mathrm{\infty }}={\sigma }_{32}^0/\mathit{\Gamma }$ in view of the fact that the coating extends vertically as | z| → ∞.

In order to demonstrate the feasibility of the present design method, we illustrate in figures 3–6 the shapes of the two locally curved interfaces L₁ and L₂ for different values of the parameters $H_2,\hspace{0.17em}p,\hspace{0.17em}{\xi }_1,\hspace{0.17em}{\xi }_2$. It is sufficient to truncate the two series in equation (3.5) at n = 100 to arrive at the highly accurate result shown in figures 3–5. By contrast, the two series in equation (3.5) should be truncated at n = 1000 to arrive at the similarly highly accurate result shown in figure 6 due to the fact that H₂ − H₁ = 0.5H₁ is relatively small. This observation implies that the rate of convergence of the sum of the two series in equation (3.5) depends on the ratio H₂/H₁. Specifically, we have rapid convergence when H₂/H₁ ≥ 2.5 (see figures 3–5) but relatively slow convergence when H₂/H₁ ≤ 1.5 (see figure 6). Countless geometries of the cloaking lamellar structure can be generated by choosing different values of the four parameters in equation (3.5).

Figure 4.

The shapes of the two interfaces for different values of ξ₁ and ξ₂ with $H_2=3H_1,\hspace{0.17em}p=2H_1$. In each subplot, the plus sign indicates the location of the dipole component with Burgers vector b while the star indicates the location of the other dipole component with Burgers vector −b. (Online version in colour.)

Figure 3.

The shapes of the two interfaces for different values of p with $H_2=3H_1,\hspace{0.17em}{\xi }_1=1.5H_1,\hspace{0.17em}{\xi }_2=2.5H_1$. In each subplot, the plus sign indicates the location of the dipole component with Burgers vector b while the star indicates the location of the other dipole component with Burgers vector −b. (Online version in colour.)

Figure 6.

The shapes of the two interfaces by choosing $H_2=1.5H_1,\hspace{0.17em}p=2H_1,\hspace{0.17em}{\xi }_1=(1.25-\text{i})H_1,\hspace{0.17em}{\xi }_2=(1.25+\text{i})H_1$. The plus sign indicates the location of the dipole component with Burgers vector b while the star indicates the location of the other dipole component with Burgers vector −b. (Online version in colour.)

Figure 5.

The shapes of the two interfaces for different values of H₂ with $p=2H_1,\hspace{0.17em}{\xi }_1=1.5H_1,\hspace{0.17em}{\xi }_2=H_2-0.5H_1$. In each subplot, the plus sign indicates the location of the dipole component with Burgers vector b while the star indicates the location of the other dipole component with Burgers vector −b. (Online version in colour.)

4. Cloaking lamellar structures for a circular Eshelby inclusion

In this section, we will design cloaking structures for a circular Eshelby inclusion undergoing either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. The circular Eshelby inclusion and its surrounding material have the same elastic constants.

(a). A circular Eshelby inclusion undergoing uniform anti-plane eigenstrains

In a similar manner, we can also design cloaking lamellar structures for a circular Eshelby inclusion undergoing uniform anti-plane eigenstrains $({\epsilon }_{31}^{\ast },{\epsilon }_{32}^{\ast })$ embedded in the middle coating as shown in figure 7. The prescribed uniform anti-plane stresses in the surrounding two half-planes having the same shear modulus are not disturbed by the circular Eshelby inclusion in the coating. The circular Eshelby inclusion has radius R with the centre located at z = z₀. Except for the Eshelby inclusion, the geometry and nomenclature are similar to those in §3. In this case, the conformal mapping function for the coating takes the form

$$\begin{array}{rl}z& =\omega (\xi )=\xi +\sum _{n=1}^{+\mathrm{\infty }}\frac{q}{\xi +{\overline{\xi }}_0-2H_1+2(n-1)(H_2-H_1)}+\sum _{n=1}^{+\mathrm{\infty }}\frac{q}{\xi +{\overline{\xi }}_0-2H_2+2(n-1)(H_1-H_2)},\\ \xi & ={\omega }^{-1}(z),H_1\le \mathrm{Re}\{\xi \}\le H_2,\end{array}$$ 4.1

where ξ₀ = ω⁻¹(z₀) or conversely z₀ = ω(ξ₀), and q is a complex number. We again mention that, as in the construction of the mapping function in equation (3.5), the construction of this particular mapping function in equation (4.1) is again inspired by the image method adopted by Chou [18] and arrived at via a careful process by considering the conditions dictated by the problem at hand.

Figure 7.

The cloaking lamellar structure for a circular Eshelby inclusion undergoing uniform anti-plane eigenstrains in the middle coating. (Online version in colour.)

As shown in figure 8, using the mapping function in equation (4.1), the middle coating including the circular Eshelby inclusion is mapped onto the strip $H_1\le \mathrm{Re}\{\xi \}\le H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the left interface L₁ is mapped onto the vertical line: $\mathrm{Re}\{\xi \}=H_1,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the right interface L₂ is mapped onto another vertical line: $\mathrm{Re}\{\xi \}=H_2,\hspace{0.17em}-\mathrm{\infty }<\text{Im}\{\xi \}<+\mathrm{\infty }$; the centre of the circular Eshelby inclusion at z = z₀ is mapped onto the point ξ = ξ₀. The mapping function in equation (4.1) contains three parameters $H_2,\hspace{0.17em}q,\hspace{0.17em}{\xi }_0$ with H₁ being a scaling constant. The two parameters H₂ and ξ₀ have the dimension of length while q has the dimension of length squared.

Figure 8.

The ξ-plane. (Online version in colour.)

Remark 2. As in remark 1, since n is taken sufficiently large, the terms appearing in each of the two individual series in equation (4.1) behave as follows:

$$\begin{array}{rl}\frac{q}{\xi +{\overline{\xi }}_0-2H_1+2(n-1)(H_2-H_1)}& \approx \frac{q}{2(n-1)(H_2-H_1)}-\frac{q(\xi +{\overline{\xi }}_0-2H_1)}{4{(n-1)}^2{(H_2-H_1)}^2},\\ \frac{q}{\xi +{\overline{\xi }}_0-2H_2+2(n-1)(H_1-H_2)}& \approx -\frac{q}{2(n-1)(H_2-H_1)}-\frac{q(\xi +{\overline{\xi }}_0-2H_2)}{4{(n-1)}^2{(H_2-H_1)}^2}.\end{array}$$

Consequently, the divergent part of each of the series in equation (4.1) ‘cancels' when summing the entire series in equation (4.1) resulting in a convergent series representing the mapping function.

In order to achieve cloaking, the analytic functions f₁(z) defined in the left half-plane and f₃(z) defined the right half-plane continue to take the form in equation (3.2) and f₂(ξ) is still of the general form in equation (3.3) with the constant C₀ being determined by equation (3.7). The following relationship can be derived

$$\begin{array}{rl}{\epsilon }_{32}^{\ast }+\text{i}{\epsilon }_{31}^{\ast }& =\frac{Cq(\mathit{\Gamma }-1)}{2R^2}\overline{{\omega }^{\mathrm{\prime }}({\xi }_0)}\\ & =\frac{C(\mathit{\Gamma }-1)}{2R^2}[q-\sum _{n=1}^{+\mathrm{\infty }}\frac{{|q|}^2}{{[{\xi }_0+{\overline{\xi }}_0-2H_1+2(n-1)(H_2-H_1)]}^2}\\ & -\sum _{n=1}^{+\mathrm{\infty }}\frac{{|q|}^2}{{[{\xi }_0+{\overline{\xi }}_0-2H_2+2(n-1)(H_1-H_2)]}^2}].\end{array}$$ 4.2

The ratio $({\epsilon }_{32}^{\ast }+\text{i}{\epsilon }_{31}^{\ast })/C$ can be determined from the above equation for given values of the parameters $H_2,\hspace{0.17em}q,\hspace{0.17em}{\xi }_0,\hspace{0.17em}\mathit{\Gamma },\hspace{0.17em}R$. We can also see from equation (4.2) that the complex nature of q comes from that of the ratio $({\epsilon }_{32}^{\ast }+\text{i}{\epsilon }_{31}^{\ast })/C$. More specifically,

$$\text{Im}\{q\}=\frac{2R^2}{\mathit{\Gamma }-1}\text{Im}\left\{\frac{{\epsilon }_{32}^{\ast }+\text{i}{\epsilon }_{31}^{\ast }}{C}\right\}.$$ 4.3

We illustrate in figures 9–11 the shapes of the two locally curved interfaces L₁ and L₂ for different values of the three parameters $H_2,\hspace{0.17em}q,\hspace{0.17em}{\xi }_0$. Interestingly, the shapes of the two interfaces in figure 9 are similar to those in figure 3. The permissible regions of H₂ and | q| for typical values of Arg(q) are illustrated in figure 12 in order to ensure a one-to-one mapping in equation (4.1) for the coating S₂. The pair (H₂, | q|) should lie just on or below the curve for a fixed value of Arg(q) in figure 12. It is seen from figure 12 that (i) when H₂ increases, the permissible range of | q| enlarges for a fixed value of Arg(q); (ii) when $|\text{Arg}(q)|,\hspace{0.17em}0\le |\text{Arg}(q)|<\frac{\pi }{2}$ increases, the permissible range of | q| enlarges for a fixed value of H₂; (iii) | q| can be arbitrary when $\text{Arg}(q)=\pm \frac{\pi }{2}$. Our numerical results further indicate that | q| can be taken to be any positive value for a given value of H₂ when $\frac{\pi }{2}\le \pm \text{Arg}(q)\le \pi$ and ξ₀ = 0.5(H₁ + H₂).

Figure 10.

The shapes of the two interfaces for different values of ξ₀ with $H_2=3H_1,\hspace{0.17em}q=-2H_1^2$. In each subplot, the star indicates the location of the centre of the circular Eshelby inclusion. (Online version in colour.)

Figure 9.

The shapes of the two interfaces for different values of q with $H_2=3H_1,\hspace{0.17em}{\xi }_0=2H_1$. In each subplot, the star indicates the location of the centre of the circular Eshelby inclusion. (Online version in colour.)

Figure 11.

The shapes of the two interfaces for different values of H₂ with $q=-2H_1^2,\hspace{0.17em}{\xi }_0=0.5(H_1+H_2)$. In each subplot, the star indicates the location of the centre of the circular Eshelby inclusion. (Online version in colour.)

Figure 12.

Condition for the mapping function in equation (4.1) to be one-to-one for the coating S₂ for different values of $\text{Arg}(q)=0,\pm \frac{\pi }{4},\pm \frac{\pi }{3},\pm \frac{\pi }{2}$ with ξ₀ = 0.5(H₁ + H₂). (Online version in colour.)

(b). A circular thermal inclusion

Next, we design cloaking lamellar structures for a circular Eshelby inclusion undergoing uniform volumetric eigenstrains: ${\epsilon }_{11}^{\ast }={\epsilon }_{22}^{\ast }={\epsilon }^{\ast },\hspace{0.17em}{\epsilon }_{12}^{\ast }=0$ in the middle coating. In this case, the Eshelby inclusion becomes a thermal inclusion [23]. The boundary of the thermal inclusion is described by: |z − z₀| = R. The presence of the circular thermal inclusion in the coating S₂ will not induce any disturbance in the prescribed uniform hydrostatic stresses in the surrounding left and right half-planes S₁ and S₃ having identical elastic properties (i.e. μ₁ = μ₃ and κ₁ = κ₃). The geometry and rules for notation are identical to those for a circular Eshelby inclusion undergoing uniform anti-plane eigenstrains.

In order to achieve cloaking in the present setting, the two pairs of analytic functions defined in the two half-planes take the form

$$\begin{array}{rl}{\phi }_1(z)& =Az,\hspace{0.17em}{\psi }_1(z)=0,z\in S_1;\\ {\phi }_3(z)& =Az+\frac{1-\mathit{\Gamma }}{\mathit{\Gamma }+{\kappa }_1}{\overline{A}}_0,\hspace{0.17em}{\psi }_3(z)=A_0,\hspace{0.17em}z\in S_3,\end{array}$$ 4.4

where A is a given real constant and A₀ is a complex constant to be determined, Γ = μ₁/μ₂ as adopted in §3. The mapping function in equation (4.1) may still be adopted here. By enforcing the continuity conditions of tractions and displacements across the two perfect interfaces L₁ and L₂, we arrive at

$${\phi }_2(\xi )={\phi }_2(\omega (\xi ))=\frac{A(2\mathit{\Gamma }+{\kappa }_1-1)}{\mathit{\Gamma }({\kappa }_2+1)}\omega (\xi ),H_1\le \mathrm{Re}\{\xi \}\le H_2;$$ 4.5

and

$$\begin{array}{rl}{\psi }_2(\xi )& ={\psi }_2(\omega (\xi ))\\ & =\frac{2A[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}\overline{\omega }(2H_1-\xi )\\ & =\frac{2A[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}\overline{\omega }(2H_2-\xi )+\frac{{\kappa }_1+1}{\mathit{\Gamma }+{\kappa }_1}{A}_0,H_1\le \mathrm{Re}\{\xi \}\le H_2.\end{array}$$ 4.6

It is quite simple to derive from equation (4.6) that

$$\overline{\omega }(2H_1-\xi )=\overline{\omega }(2H_2-\xi )+\frac{A_0\mathit{\Gamma }({\kappa }_1+1)({\kappa }_2+1)}{2A(\mathit{\Gamma }+{\kappa }_1)[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]},H_1\le \mathrm{Re}\{\xi \}\le H_2,$$ 4.7

the structure of which is similar to that of equation (3.4).

The complex constant A₀ can be uniquely determined from equations (4.1) and (4.7) as

$$A_0=\frac{4A(H_1-H_2)(\mathit{\Gamma }+{\kappa }_1)[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_1+1)({\kappa }_2+1)},$$ 4.8

which implies that A₀ is real-valued.

It is deduced from equations (4.5) and (4.6) that

$${\phi }_2(z)\cong \frac{A(2\mathit{\Gamma }+{\kappa }_1-1)}{\mathit{\Gamma }({\kappa }_2+1)}z+O(1),\hspace{0.17em}{\psi }_2(z)\cong -\frac{2A[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}z+O(1),|z|\to \mathrm{\infty }.$$ 4.9

Thus, when the prescribed uniform hydrostatic stresses in the two half-planes are given by

$${\sigma }_{11}={\sigma }_{22}={\sigma }_0=2A,\hspace{0.17em}{\sigma }_{12}=0,\hspace{0.17em}z\in S_1\cup S_3,$$ 4.10

the stress state at infinity in the middle coating is determined from equations (2.1) and (4.9) as follows

$${\sigma }_{11}\cong {\sigma }_0,\hspace{0.17em}{\sigma }_{22}\cong \frac{{\sigma }_0[\mathit{\Gamma }(3-{\kappa }_2)+2({\kappa }_1-1)]}{\mathit{\Gamma }({\kappa }_2+1)},\hspace{0.17em}{\sigma }_{12}\cong 0,|z|\to \mathrm{\infty }.$$ 4.11

We see from the above that the coating should be subjected to the following uniform remote loading:

$${\sigma }_{22}^{\mathrm{\infty }}=\frac{{\sigma }_0[\mathit{\Gamma }(3-{\kappa }_2)+2({\kappa }_1-1)]}{\mathit{\Gamma }({\kappa }_2+1)}.$$ 4.12

It is seen from equations (4.5) and (4.10) that the mean stress σ₁₁ + σ₂₂ is uniformly distributed in $S_2^{\mathrm{\prime }}$ denoting the middle coating outside the circular thermal inclusion as follows

$${\sigma }_{11}+{\sigma }_{22}=\frac{2{\sigma }_0(2\mathit{\Gamma }+{\kappa }_1-1)}{\mathit{\Gamma }({\kappa }_2+1)},\hspace{0.17em}z\in S_2^{\mathrm{\prime }}.$$ 4.13

It is further derived from equations (4.10) and (4.13) that the hoop stress is constant along the two interfaces L₁ and L₂ on the coating side, and is given by

$${\sigma }_{tt}=\frac{{\sigma }_0[\mathit{\Gamma }(3-{\kappa }_2)+2({\kappa }_1-1)]}{\mathit{\Gamma }({\kappa }_2+1)},\hspace{0.17em}z\in (L_1\cup L_2)\cap S_2,$$ 4.14

which indicates that the design criterion of ‘equal strength' advanced by Cherepanov [25] has also been satisfied. Note that the expression of σ_tt in equation (4.14) and that of ${\sigma }_{22}^{\mathrm{\infty }}$ in equation (4.12) are identical.

In addition, the following relationship can be derived

$$\frac{{\mu }_1{\epsilon }^{\ast }}{{\sigma }_0}=\frac{\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1}{4R^2}q\overline{{\omega }^{\mathrm{\prime }}({\xi }_0)}=\frac{\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1}{4R^2}(q-{|q|}^{2}Y),$$ 4.15

where

$$\begin{array}{rl}Y& =\overline{Y}=\sum _{n=1}^{+\mathrm{\infty }}\frac{1}{{[{\xi }_0+{\overline{\xi }}_0-2H_1+2(n-1)(H_2-H_1)]}^2}+\sum _{n=1}^{+\mathrm{\infty }}\frac{1}{{[{\xi }_0+{\overline{\xi }}_0-2H_2+2(n-1)(H_1-H_2)]}^2}>0.\end{array}$$ 4.16

It is clearly seen from equation (4.15) that the parameter q must be real-valued. The ratio μ₁ε*/σ₀ can be simply determined once the positive parameter Y in equation (4.16) is calculated. The variation of Y as a function of $\mathrm{Re}\{{\xi }_0\}$ for different values of H₂ is illustrated in figure 13. It is seen from figure 13 that Y attains its minimum when $\mathrm{Re}\{{\xi }_0\}=0.5(H_1+H_2)$ and becomes infinite as $\mathrm{Re}\{{\xi }_0\}\to H_1$ or $\mathrm{Re}\{{\xi }_0\}\to H_2$. The permissible regions of $\mathrm{Re}\{{\xi }_0\}$ and q for different values of H₂ are illustrated in figure. 14 in order to ensure a one-to-one mapping in equation (4.1) for the coating S₂. The pair $[(\mathrm{Re}\{{\xi }_0\}-H_1)/(H_2-H_1),\hspace{0.17em}q/H_1^2]$ should lie just on or below the curve for a fixed value of H₂/H₁ in figure 14. It is stressed that q can be taken as any negative value. Our extensive calculations suggest that the term 1 − qY is always positive. This fact implies that the sign of the ratio μ₁ε*/σ₀ is the same as that of the product q[Γ(κ₂ − 1) − κ₁ + 1].

Figure 13.

The variation of Y as a function of $\mathrm{Re}\{{\xi }_0\}$ for different values of H₂. (Online version in colour.)

Figure 14.

Condition for the mapping function in equation (4.1) to be one-to-one for the coating S₂ for different values of H₂. (Online version in colour.)

5. Cloaking lamellar structures for a concentrated couple

Finally, as a consequence, we will design cloaking lamellar structures for a concentrated couple with the moment M located at z = z₀ in the middle coating. The complex potentials induced by a concentrated couple in a homogeneous elastic plane can be found in Suo [24]. In the current setting, we continue to adopt the conformal mapping function in equation (4.1) and equations (4.4)–(4.14) continue to remain valid. However, the relationship in equation (4.15) should be replaced by the following

$$\frac{\text{i}M}{2\pi {\sigma }_0}=\frac{[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}q\overline{{\omega }^{\mathrm{\prime }}({\xi }_0)}=\frac{[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}(q-{|q|}^{2}Y),$$ 5.1

where Y has been defined in equation (4.16). We further denote by q′ and q′′ the real and imaginary parts of q, respectively.

From equation (5.1), we find that

$$\frac{M}{2\pi {\sigma }_0}=\frac{q^{\mathrm{\prime }\mathrm{\prime }}[\mathit{\Gamma }({\kappa }_2-1)-{\kappa }_1+1]}{\mathit{\Gamma }({\kappa }_2+1)}.$$ 5.2

The relationship in equation (5.2) implies that once q′′ and the material parameters $\mathit{\Gamma },\hspace{0.17em}{\kappa }_1,\hspace{0.17em}{\kappa }_2$ for the composite are given, the term on the left-hand side of equation (5.2) will be determined. Once q′′ is given, the following quadratic equation in q′ can be derived from equation (5.1)

$$Y{q}^{\mathrm{\prime }2}-q^{\mathrm{\prime }}+Y{q}^{\mathrm{\prime }\mathrm{\prime }2}=0.$$ 5.3

Thus, q′ can be determined from the above as

$$q^{\mathrm{\prime }}=\frac{1\pm \sqrt{1-4{(Y{q}^{\mathrm{\prime }\mathrm{\prime }})}^2}}{2Y},|q^{\mathrm{\prime }\mathrm{\prime }}|\le \frac{1}{2Y}.$$ 5.4

It is seen from the above analysis that in order to achieve cloaking structures for a concentrated couple in the middle coating, both the real and imaginary parts of the parameter q should be non-zero. For example, when we choose $H_2=4H_1,\hspace{0.17em}{\xi }_0=0.5(H_1+H_2)=2.5H_1$ and $q^{\mathrm{\prime }\mathrm{\prime }}=1.5H_1^2$, we find from equation (4.16) that $Y=0.2736H_1^{-2}$. Consequently, we determine from equation (5.4) that $q^{\mathrm{\prime }}=2.8714H_1^2,\hspace{0.17em}0.7836H_1^2$. It is impossible to arrive at a physically permissible geometry using the solution $q=(2.8714+1.5\text{i})H_1^2$. On the other hand, by using the other solution $q=(0.7836+1.5\text{i})H_1^2$, we can obtain a permissible geometry for the composite as illustrated in figure 15. In all our calculations for different values of the three given parameters $H_2,\hspace{0.17em}{\xi }_0,\hspace{0.17em}{q}^{\mathrm{\prime }\mathrm{\prime }}$, taking the positive sign in equation (5.4) cannot lead to an admissible solution.

Figure 15.

The shapes of the two interfaces when choosing $H_2=4H_1,\hspace{0.17em}{\xi }_0=2.5H_1,\hspace{0.17em}q=(0.7836+1.5\text{i})H_1^2$. The star indicates the location of the concentrated couple at z = z₀. (Online version in colour.)

6. Conclusion

Through satisfaction of the continuity conditions across the two perfect interfaces L₁ and L₂, we have constructed the mapping function in equation (3.5) for the middle coating in order to achieve the design goal that the prescribed uniform anti-plane stress fields in both the left and right half-planes are not disturbed in the presence of the screw dislocation dipole. Actually, the present method can be conveniently modified to incorporate the more general situation in which there exist an arbitrary number of screw dislocation dipoles in the coating. We have also designed cloaking lamellar structures for a circular Eshelby inclusion and for a concentrated couple with the corresponding mapping function given by equation (4.1). In the context of steady-state heat conduction, the cloaking lamellar structures obtained in §3 can be interpreted as follows: the disturbance in temperature generated by a line heat source and another line heat sink in the coating is confined only to the coating and there is no disturbance in the original linear temperature distributions in the surrounding two half-planes resulting from the prescribed uniform heat flux. The cloaking lamellar structures for a circular Eshelby inclusion undergoing uniform anti-plane eigenstrains obtained in §4a can also be conveniently interpreted in terms of steady-state heat conduction [26].

We end by noting that further research is required to generalize the present theory to account for more general material models. For example, in [27] it is shown that the presence of concentrated forces or moments necessitate a generalization of standard Cauchy material models to include non-local effects, for example, to material models incorporating strain gradient theory [28] or micromorphic theory [29].

Appendix A

The continuity conditions of traction and displacement across the two perfect interfaces L₁ and L₂ can be expressed in terms of the three analytic functions $f_1(z),\hspace{0.17em}{f}_2(z)$ and f₃(z) as follows

$$\begin{array}{rl}{f}_2(z)+\overline{f_2(z)}& =\mathit{\Gamma }{f}_1(z)+\mathit{\Gamma }\overline{f_1(z)},\\ f_2(z)-\overline{f_2(z)}& =f_1(z)-\overline{f_1(z)},z\in L_1;\end{array}$$ A 1

and

$$\begin{array}{rl}{f}_2(z)+\overline{f_2(z)}& =\mathit{\Gamma }{f}_3(z)+\mathit{\Gamma }\overline{f_3(z)},\\ f_2(z)-\overline{f_2(z)}& =f_3(z)-\overline{f_3(z)},z\in L_2.\end{array}$$ A 2

By using equation (3.2) and the technique of analytic continuation [6,7,23,24], we have from equation (A 1) that

$$f_2(\xi )=\frac{C(\mathit{\Gamma }+1)}{2}\omega (\xi )+\frac{\overline{C}(\mathit{\Gamma }-1)}{2}\overline{\omega }(2H_1-\xi ),\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2,$$ A 3

and meanwhile we have from equation (A 2) that

$$f_2(\xi )=\frac{C(\mathit{\Gamma }+1)}{2}\omega (\xi )+\frac{\overline{C}(\mathit{\Gamma }-1)}{2}\overline{\omega }(2H_2-\xi )+\frac{C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)}{2},\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2.$$ A 4

Equation (3.3) can be obtained from equations (A 3) and (A 4) as a consequence of the consistency condition for f₂(ξ).

Equation (3.4) is equivalent to

$$\overline{\omega }(2H_1-\xi )-\overline{\omega }(2H_2-\xi )=\frac{C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)}{\overline{C}(\mathit{\Gamma }-1)},\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2.$$ A 5

The mapping function (3.5) is developed in a sequence of attempts as follows. At first, we try the following mapping function

$$z=\omega (\xi )=\xi +p\sum _{n=1}^{+\mathrm{\infty }}\ln \frac{\xi +{\overline{\xi }}_1-2H_1+2(n-1)(H_2-H_1)}{\xi +{\overline{\xi }}_2-2H_1+2(n-1)(H_2-H_1)},\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2,$$ A 6

which, when substituted into equation (A 5) yields

$$2(H_1-H_2)+\overline{p}\ln \frac{\xi -{\xi }_1}{\xi -{\xi }_2}=\frac{C_0(\mathit{\Gamma }+1)+{\overline{C}}_0(\mathit{\Gamma }-1)}{\overline{C}(\mathit{\Gamma }-1)},\hspace{0.17em}{H}_1\le \mathrm{Re}\{\xi \}\le H_2.$$ A 7

which contains an unwanted additional logarithmic function on the left-hand side. In order to remove this logarithmic function, our next attempt is the mapping function in equation (3.5), in which the second series can be deduced quite simply from the first series by replacing H₁ with H₂ and replacing H₂ with H₁ in the first series. It can be verified quite simply that upon substitution of equation (3.5) into equation (A 5), the logarithmic function on the left-hand side of equation (A 7) can be completely removed due to the addition of the second series in equation (3.5) and meanwhile there is no additional term in the final expression. This fact implies that the choice of equation (3.5) is indeed successful for our purposes.

Data accessibility

Provided as supplementary material.

Authors' contributions

X.W. conceived the problem, performed the theoretical and numerical analysis and drafted the manuscript; P.S. contributed to the mathematics, critically revised the manuscript coordinated the study and helped draft the manuscript. Both authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests

Funding

This work is supported by the National Natural Science Foundation of China (grant no. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN – 2017 - 03716115112).