Does the capacitor analogy model in fracture mechanics of elastic dielectrics constitute an appropriate approximation?

Abstract

The analytical solution of an elliptic dielectric cavity in an infinite dielectric plate is taken as basis to investigate a Griffith crack problem which is obtained by letting the semi-minor axis tend towards zero. In the course of this, an erroneous conformal mapping, commonly employed in literature and correctly reproducing the electric field only in a part of the physical space, is rectified. Interpreting the elliptic interface as faces of a mechanically opened crack which is exposed to an oblique remote electric field, surface charges and electrostatic tractions are calculated. In contrast to the established capacitor analogy model, approximately yielding electric charge densities and Coulombic tractions from displacements and electric potentials in the undeformed crack configuration, the work at hand provides exact solutions accounting for different implications of the crack curvature and for the inclination of the electric field. Crack weight functions are finally used to calculate stress and electric displacement intensity factors. As turns out, the surface charges of the capacitor analogy represent an excellent substitute for the exact electric boundary conditions within a relevant range of parameters, whereas inaccurate Coulombic tractions in the vicinity of crack tips may lead to a significantly overestimated mode I stress intensity factor.

1. Introduction

In fracture mechanics of dielectrics and piezoelectrics, electromechanical boundary value problems of cracked bodies under mixed loading conditions are considered in general. Within infinitesimal strain theory, being appropriate in the case of brittle material, current and initial configurations are not distinguished, allowing for a problem formulation with respect to the latter. Several established methods for the analytical and numerical solution of these problems are commonly pursued in literature, where the electromechanical coupling and material anisotropy represent aggravating circumstances. On the one hand, analytically, integral transformations [1–3] or Green’s functions [4–6] are oftentimes exploited. In the case of two-dimensional problems, complex analysis provides a rich framework from which the powerful Stroh formalism [7–9] and extended Lekhnitskii theory [10, 11] emanate. Moreover, by conformally mapping complicated to simple geometries, solutions may be elegantly adjusted to values prescribed on arbitrary boundaries [10, 12, 13]. On the other hand, the finite element method [14–18] and the boundary element method [6, 19–22] are predominant in the field of numerical studies, where the latter is based on Green’s functions in infinite domains and, upon dispensing with volume discretization, for crack problems oftentimes turns out superior to the former in terms of computational efficiency.

When dealing with sharp cracks in linear elastic fracture mechanics disregarding cohesive zones as well as internal pressure, the crack faces represent Neumann boundaries, where zero tractions are prescribed. The boundary conditions on the crack faces in a dielectric medium, however, are less unambiguous since upon loading, a crack medium which is either conductive or of permittivity ${\kappa }^{\mathrm{C}}$ , oftentimes air or vacuum, will fill the deformed crack. Due to the relevance of most technical applications, the focus of this work is on dielectric rather than on conducting crack media.

In order to properly account for this condition in the electrostatic boundary value problem, both solid body and crack medium have to be subjected to Maxwell’s equations as well as appropriate interface conditions on the crack faces. Circumventing this effort, researchers have deduced two special cases of boundary conditions from the general two-domain problem [23]. The impermeable crack model, initially introduced in Deeg [24] and Pak [25], employs ${\kappa }^{\mathrm{C}}=0$ , which, resulting in a vanishing surface charge density prescribed as homogeneous Neumann boundary conditions on the crack faces, represents a straightforward generalization of the traction-free conditions. The field lines of the electric displacement thus entirely circumvent the crack. The permeable model tracing back to Parton [1], on the other hand, stipulates zero voltage and continuity of the normal component of the electric displacement across two opposite points on the crack faces, which follows from either ${\kappa }^{\mathrm{C}}\to \infty$ or a closed crack with ${\kappa }^{\mathrm{C}}>0$ [26]. In this case, electric field lines are undisturbed by the presence of the crack. As shown in the literature [3, 13, 27, 28], these two extreme models in general poorly reflect the real electric boundary conditions and may, at best, provide lower and upper bounds for the surface charge density. Consequently, the dielectric intensity factor (DIF), characterizing the electric singularity at the crack tip, may be significantly underestimated or overestimated. Notwithstanding, owing to their simplicity, both models are oftentimes used in literature [4, 10, 22, 29–36].

In an attempt to capture the circumstances of a crack medium with finite permittivity ${\kappa }^{\mathrm{C}}$ and crack face displacement, Hao and Shen [37] (and earlier Parton and Kudryavtsev [38]) proposed the so-called semi-permeable model, which considers pairwise opposite points on the crack faces as one-dimensional plate capacitors. The validity of this ansatz, also referred to as capacitor analogy, is predicated on infinitely small crack openings, preserving the initial configuration and thus the straightness of crack faces. By exploiting standard capacitor equations, the electric displacement inside the crack becomes a function of the unknown local jumps of electric potential and displacement, which allows for the prescription of approximate surface charges on the undeformed crack faces, retaining the infinitesimal strain framework. As a consequence, the DIF becomes particularly sensitive to the permittivity ${\kappa }^{\mathrm{C}}$ . Due to these nonlinear Neumann boundary conditions, problems are typically solved iteratively. The capacitor analogy is furthermore consistent with the two limiting cases of permeable and impermeable conditions.

Extending the capacitor analogy and building on the work of Kemmer and Balke [39], Landis and McMeeking [40] introduced Coulombic tractions acting on the crack faces. These tractions are obtained from the electric field and electric displacement inside the crack and represent a simplification of electric surface tractions due to Maxwell stress. The latter has been controversially discussed by renowned physicists in the past [41, 42] and, besides tractions, gives rise to body forces and couples induced by electric and magnetic fields [43]. Unfortunately, several formulations of the Maxwell stress tensor are known nowadays, however, none could be validated without contradiction to date. In dielectric and piezoelectric crack models, tractions emanating from a simplification of the approach according to Minkowski prevail. The closing effect of the Coulombic tractions on various crack configurations has been extensively investigated by several researchers [20, 21, 44–49], who generally recorded a partially considerable reduction of the mode I stress intensity factor (SIF). Altogether, the capacitor analogy, sometimes complemented by the Coulombic tractions, is commonly applied in corresponding literature of both analytical and numerical nature [16,18,44–53].

Notwithstanding, this model exhibits some fundamental deficiencies in the context of considering finite deformation of a crack within a geometrically linear framework of solid mechanics. Consequently, the electrostatic potential used to calculate the electric field inside the crack is determined in the undeformed configuration with a closed slit crack. Furthermore, the model presupposes an electric field inside the crack which is orthogonal to the crack faces, neglecting possible tangential components at the crack tip or generally due to inclined electrical loading. Finally, when applying Coulombic tractions and surface charges on the crack faces, the unit normal of the undeformed configuration is used, disregarding curvature of the distorted crack, in particular being relevant near the crack tips. While a few works numerically investigate the errors made due to some of the above-mentioned aspects, seemingly validating at least the surface charge approximation even for inclined electric loads [54, 55], a comprehensive validation and assessment of errors based on an analytical solution of the dielectric two-domain problem is still pending. The influence of crack curvature on induced electrostatic tractions in their complete formulation and its implications on crack tip loading within a fracture mechanics framework has not yet been assessed, so the validity of the assumptions made for the Coulombic tractions remains to be evaluated in principle.

In this work, these shortcomings are covered, as the approximate surface charges and Coulombic tractions according to the capacitor analogy are checked against the corresponding true charges and tractions prevailing at a deformed crack of elliptic shape in an infinite linear-elastic isotropic dielectric plate subjected to remote electric fields and tensile stresses. Deviations of stress and dielectric intensity factors evolving from approximate surface charges and reactions, on the one hand, and true ones, on the other hand, are finally evaluated based on crack weight functions. As a result of the constitutive assumptions, electrical and mechanical boundary value problems decouple, so that the former may be solved independently of the latter. Body forces and couples due to some formulations of Maxwell stress tensors, entering the mechanical problem, are shortly outlined, however, disregarded in the calculations. The electrostatic problem has been investigated and solved by numerous authors exploiting complex analysis and conformal mapping [12,56–58]. However, some mathematical subtleties previously gone unnoticed will be pointed out in the solution procedure rigorously derived in the following. These include the proof of the existence of a holomorphic potential for the electric field on a multiply connected domain, where holomorphy alone does not suffice, as well as the complex differentiability of said potential at the slit between the ellipse’s focal points, which is usually tacitly assumed but indeed follows from the requirement of continuity. What is more, an erroneous conformal mapping commonly used in literature is rectified, having consequences for a broad range of similar mappings involved in the more general context of anisotropic elastic or piezoelectric problems, where the correct solution could hitherto only be obtained in certain subdomains of the physical plane.

2. Theoretical framework

2.1. Infinite dielectric with elliptic cavity

Let a linear isotropic dielectric body with permittivity ${\kappa }^{\mathrm{o}}\ge {\kappa }_0$ occupy the infinite three-dimensional Euclidean space minus an elliptic cylinder with semi-major and semi-minor axes $a$ and $b$ , respectively, of infinite height placed at the origin. The cylinder is filled with another linear isotropic dielectric or vacuum with permittivity ${\kappa }^{\mathrm{i}}\ge 0$ . While the interior of both regions is governed by the electrostatic Maxwell equations in terms of Gauss’ and Faraday’s law in the absence of volume charges, the inner material boundary surface devoid of unbounded charge is subjected to the corresponding interface conditions requiring the continuity of tangential and normal components of the electric field and electric displacement, respectively. At infinity, a constant electric field is prescribed in the $x_1$ - $x_2$ plane, which, considering the problem’s symmetry, results in vanishing $x_3$ -components and zero gradients of the electric field and displacement vectors along the $x_3$ -axis. This allows for the reduction of the problem to two dimensions. Letting

$${\mathrm{\Omega }}^{\mathrm{i}}=\{\mathbf{x}\in {\mathbb{R}}^2|{(x_1/a)}^2+{(x_2/b)}^2<1\}$$

(1)

denote the set of all points lying inside the ellipse $\mathrm{\Gamma }=\partial {\mathrm{\Omega }}^{\mathrm{i}}$ , and thus ${\mathrm{\Omega }}^{\mathrm{o}}={\mathbb{R}}^2\backslash \overline{{\mathrm{\Omega }}^{\mathrm{i}}}$ refer to the set of all points lying outside, the constitutive equation interlinking electric displacement $\mathbf{D}$ and electric field $\mathbf{E}$ reads

$$\mathbf{D}=\{\begin{array}{c}{\kappa }^{\mathrm{o}}\mathbf{E}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{o}},\hfill \\ \multicolumn{1}{c}{{\kappa }^{\mathrm{i}}\mathbf{E}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{i}}.}\end{array}$$

(2)

In conclusion, the following boundary value problem emerges: the vector fields $\mathbf{E},\mathbf{D}\in C^1({\mathrm{\Omega }}^{\mathrm{i}}\cup {\mathrm{\Omega }}^o,{\mathbb{R}}^2)$ are sought, so that the field equations

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{div}\mathbf{D}=0,}\\ \multicolumn{1}{c}{\mathrm{curl}\mathbf{E}=0,}\end{array}$$

(3)

in ${\mathrm{\Omega }}^{\mathrm{i}}\cup {\mathrm{\Omega }}^{\mathrm{o}}$ , the interface conditions

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{e}}^{\mathrm{t}}·[[\mathbf{E}]]=0,}\\ \multicolumn{1}{c}{{\mathbf{e}}^{\mathrm{n}}·[[\mathbf{D}]]=0,}\end{array}$$

(4)

on $\mathrm{\Gamma }$ , as well as the condition at infinity

$$\mathbf{E}(\mathbf{x})\to {\mathbf{E}}^{\infty }=\mathrm{const},|\mathbf{x}|\to \infty$$

(5)

are satisfied. In equation (4), $\mathrm{\Gamma }$ is parameterized counterclockwise and ${\mathbf{e}}^{\mathrm{t}}$ and ${\mathbf{e}}^{\mathrm{n}}$ represent unit vectors tangential and normal to $\mathrm{\Gamma }$ , respectively, where ${\mathbf{e}}^{\mathrm{n}}$ points outward. Jump brackets $[[·]]$ indicate the difference between the embraced quantity’s limit values approaching from either domain, e.g.,

$$[[\mathbf{E}]](\mathbf{x})=\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{\mathrm{o}}\to \mathbf{x}}{{\mathbf{x}}^{\mathrm{o}}\in {\mathrm{\Omega }}^{\mathrm{o}}}}{lim}\mathbf{E}({\mathbf{x}}^o)-\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{\mathrm{i}}\to \mathbf{x}}{{\mathbf{x}}^{\mathrm{i}}\in {\mathrm{\Omega }}^{\mathrm{i}}}}{lim}\mathbf{E}({\mathbf{x}}^i),\mathbf{x}\in \mathrm{\Gamma }.$$

(6)

${\mathbf{E}}^{\infty }$ in equation (5) is the constant remote electric field. For the sake of clarity, the fields are split with respect to the domains in terms of

$$\mathbf{E}=\{\begin{array}{c}{\mathbf{E}}^{\mathrm{o}}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{o}},\hfill \\ \multicolumn{1}{c}{{\mathbf{E}}^{\mathrm{i}}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{i}},}\end{array}\mathbf{D}=\{\begin{array}{c}{\mathbf{D}}^{\mathrm{o}}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{o}},\hfill \\ \multicolumn{1}{c}{{\mathbf{D}}^{\mathrm{i}}\mathrm{in}{\mathrm{\Omega }}^{\mathrm{i}},}\end{array}$$

(7)

where ${\mathbf{E}}^{\mathrm{o}},{\mathbf{D}}^{\mathrm{o}}\in C^1(\overline{{\mathrm{\Omega }}^{\mathrm{o}}},{\mathbb{R}}^2)$ and ${\mathbf{E}}^{\mathrm{i}},{\mathbf{D}}^{\mathrm{i}}\in C^1(\overline{{\mathrm{\Omega }}^{\mathrm{i}}},{\mathbb{R}}^2)$ . The problem outlined so far is depicted in Figure 1.

Figure 1.

Illustration of the boundary value problem of an infinite isotropic dielectric with elliptic cavity subjected to remote electric loads.

The following solution proceeds in the style of previous studies [12,59,60], whose authors first successfully exploited complex analysis to deal with the problem at hand. Accounting for equation (2) in equation (3), while bearing in mind ${\kappa }^{\mathrm{o}},{\kappa }^{\mathrm{i}}=\mathrm{const}$ , results in the two differential equations

$$\frac{\partial E_1}{\partial x_1}=-\frac{\partial E_2}{\partial x_2},$$

(8)

$$\frac{\partial E_1}{\partial x_2}=\frac{\partial E_2}{\partial x_1},$$

(9)

for the Cartesian components of $\mathbf{E}$ . Letting $z=x_1+i{x}_2$ denote a complex variable, the complex representation of the electric field

$$E(z)=E_1(x_1,x_2)+\mathrm{i}{E}_2(x_1,x_2)$$

(10)

is introduced. With equation (10) available, equations (8) and (9) disclose themselves as the Cauchy-Riemann equations for $\overline{E(z)}$ , whose complex differentiability in ${\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}\cup {\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{o}}$ follows immediately. What is more, the global formulation of Gauss’ and Faraday’s law, from which equations (3) and (4) emanate, require for any simple closed path $\gamma$ in ${\mathbb{R}}^2$ that

$${\int }_{\gamma }\mathbf{E}·{\mathbf{e}}^{\mathrm{t}}\mathrm{d}s=0,$$

(11)

$${\int }_{\gamma }\mathbf{D}·{\mathbf{e}}^{\mathrm{n}}\mathrm{d}s=0.$$

(12)

Validity of equations (11) and (12) for arbitrary piecewise continuously differentiable closed paths in ${\mathrm{\Omega }}^{\mathrm{i}}$ and ${\mathrm{\Omega }}^{\mathrm{o}}$ , i.e., regions in which $\mathbf{E}$ and $\mathbf{D}$ are continuous, ensures the existence of a complex antiderivative of $\overline{E(z)}$ in ${\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}\cup {\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{o}}$ . This fact allows us to obtain the electric field as complex derivative of a holomorphic potential $w\in \mathcal{H}({\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}\cup {\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{o}})$ , i.e.,

$$\overline{E(z)}=-w^{\prime }(z)\mathrm{in}{\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}\cup {\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{o}},$$

(13)

where $({)}^{\prime }$ denotes differentiation with respect to the corresponding argument, in this case with respect to $z$ . The potential itself can be written as

$$w(z)=\alpha (x_1,x_2)+\mathrm{i}\beta (x_1,x_2).$$

(14)

Equations (8)–(14) convey the physical interpretation of its real and imaginary parts $\alpha ,\beta \in C^{\infty }({\mathrm{\Omega }}^{\mathrm{i}}\cup {\mathrm{\Omega }}^{\mathrm{o}},\mathbb{R})$ as electrostatic potential and stream function, respectively, which relate to the electric field vector by

$$\mathbf{E}=-\mathrm{grad}\alpha =-\frac{\partial \beta }{\partial x_2}{\mathbf{e}}_1+\frac{\partial \beta }{\partial x_1}{\mathbf{e}}_2.$$

(15)

Accounting for equation (15), it becomes clear that even $\mathbf{E}\in C^{\infty }({\mathrm{\Omega }}^{\mathrm{i}}\cup {\mathrm{\Omega }}^{\mathrm{o}},{\mathbb{R}}^2)$ holds. Through transformation of the interface conditions (equation (4)) for $\mathbf{E}$ and $\mathbf{D}$ to those involving $\alpha$ and $\beta$ , an equivalent formulation related to the complex potential,

$$\begin{array}{c}\multicolumn{1}{c}{w^{\mathrm{o}}+\overline{w^{\mathrm{o}}}=w^{\mathrm{i}}+\overline{w^{\mathrm{i}}},}\\ \multicolumn{1}{c}{{\kappa }^{\mathrm{o}}[\overline{w^{\mathrm{o}}}-w^{\mathrm{o}}]={\kappa }^{\mathrm{i}}[\overline{w^{\mathrm{i}}}-w^{\mathrm{i}}],}\end{array}$$

(16)

on ${\mathrm{\Gamma }}_{\mathbb{C}}$ is obtained, which is split between the two domains analogously to equation (7), and where $w^{\mathrm{o}}$ and $w^{\mathrm{i}}$ are presumed holomorphic in a neighborhood of $\overline{{\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{o}}}$ and $\overline{{\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}}$ , respectively. Since equation (16) is not commonly used in the literature, it is derived in detail in Appendix 4 Finally, in light of equations (5) and (13), the condition at infinity reads

$$w^{\prime }(z)\to -\overline{E^{\infty }}=\mathrm{const},|z|\to \infty .$$

(17)

The problem has been simplified, so far, to identifying a scalar holomorphic potential which satisfies the corresponding conditions at the interface and infinity, equations (16) and (17), respectively. Notwithstanding, the elliptic boundary shape still complicates the solution procedure. In order to circumvent this inconvenience, consider the following conformal map ¹ $g$ with ellipse parameters $a=R(1+m),b=R(1-m)$ and the complex variable $\zeta$ :

$$\begin{array}{c}\multicolumn{1}{c}{g:D(g)\to \mathbb{C}\backslash [-2R\sqrt{m},2R\sqrt{m}],}\\ \multicolumn{1}{c}{\zeta \mapsto g(\zeta )=R(\zeta +m{\zeta }^{-1}),R>0,m\in [0,1),}\end{array}$$

(18)

where $D(g)=\mathbb{C}\backslash \overline{\mathbb{D}(\sqrt{m})}$ and the codomain excludes the line segment between the ellipse’s focal points $\pm 2R\sqrt{m}$ . By means of equation (18), points on the complex unit circle ${\mathbb{S}}^1$ are mapped onto points on the complex ellipse ${\mathrm{\Gamma }}_{\mathbb{C}}$ , which becomes obvious, e.g., by inserting the parameterization $\zeta =e^{\mathrm{i}\vartheta }$ with $\vartheta \in [0,2\pi ]$ into equation (18):

$$g(e^{\mathrm{i}\vartheta })=R(1+m)\cos (\vartheta )+\mathrm{i}R(1-m)\sin (\vartheta ).$$

(19)

It is easily verified that in the special cases of $m=0$ and $m=1$ , the ellipse degenerates into a circle and a slit crack of diameter and length $4R$ , respectively. Instead of the previously defined $D(g)$ , the domain $\mathbb{D}(\sqrt{m})\backslash \{0\}$ would be likewise suitable for equation (18). The mapping’s confinement to either domain, however, is necessary for $g$ to be a bijective function possessing an inverse, which is shown in Appendix 4, where $g$ ’s bijectivity is proven in detail. Therefore, only the domain $D(g)$ will be of interest hereinafter, which maps the annulus $D(g)\cap \mathbb{D}(1)$ and the exterior of the unit circle $D(g)\backslash \overline{\mathbb{D}(1)}$ in the $\zeta$ -plane onto the interior and exterior, respectively, of the ellipse ${\mathrm{\Gamma }}_{\mathbb{C}}$ in the $z$ -plane. The inverse function $g^{-1}:\mathbb{C}\backslash [-2R\sqrt{m},2R\sqrt{m}]\to D(g)$ , on the other hand, maps back from the $z$ - to the $\zeta$ -plane. Recapitulating, the transformation behavior of $g$ and $g^{-1}$ is illustrated in Figure 2.

Figure 2.

Transformation from the $\zeta$ - to the $z$ -plane and vice versa by means of the conformal maps $g(\zeta )$ and $g^{-1}(z)$ .

The inverse function is determined by solving $z=g(\zeta )$ for $\zeta \ne 0$ :

$$\begin{array}{c}\multicolumn{1}{c}{z=g(\zeta )=R\zeta +\mathrm{Rm}{\zeta }^{-1}=\frac{R{\zeta }^2+\mathrm{Rm}}{\zeta }}\\ \multicolumn{1}{c}{\iff 0={\zeta }^2-\frac{z}{R}\zeta +m.}\end{array}$$

(20)

Upon application of the quadratric formula, two possible solutions ${\zeta }_{1,2}(z)$ are identified in terms of

$${\zeta }_{1,2}(z)=\frac{z}{2R}\pm \sqrt{\frac{z^2}{4R^2}-m}=\frac{1}{2R}(z\pm \sqrt{z^2-4m{R}^2}).$$

(21)

As is tacitly assumed if not stated otherwise, the square root of a complex number is meant to represent the principal value with a branch cut along the negative real axis. In this case, the square root in equation (21) is discontinuous on $z\in (-2R\sqrt{m},2R\sqrt{m})$ as well as the imaginary axis, where the latter discontinuity lying in the interior of the codomain disqualifies both ${\zeta }_1$ and ${\zeta }_2$ from being appropriate inverse functions of $g$ . By means of the derivation given in Appendix 4, the inverse function reads instead

$$g^{-1}(z)=\frac{1}{2R}(z+\sqrt{z+2R\sqrt{m}}\sqrt{z-2R\sqrt{m}}),$$

(22)

where the product of the two square roots is holomorphic on $\mathbb{C}\backslash [-2R\sqrt{m},2R\sqrt{m}]$ , cf. [61]. Equation (22) is also consistent with equation (21) due to $g^{-1}(z)\in \{{\zeta }_1(z),{\zeta }_2(z)\}$ for all $z\in \mathbb{C}$ . In the literature dealing with two-dimensional problems of elasto- and electrostatics [28,56,57,62], the inverse function is oftentimes given by the incorrect expression

$${\zeta }_1(z)=\frac{1}{2R}(z+\sqrt{z^2-4m{R}^2}),$$

(23)

where the solution ${\zeta }_2$ is furthermore ignored. In Meng and Gao [58], the inverse is based on Heaviside functions choosing the solution from equation (21) which lies in $\mathbb{C}\backslash \mathbb{D}(\sqrt{m})$ , however, potentially leading to incorrect values when approaching the slit $[-2R\sqrt{m},2R\sqrt{m}]$ . A second solution is likewise neglected in the more complicated case where $R$ and $m$ are replaced by complex constants [10,13,63–65], the issue with the square root’s discontinuity prevailing nevertheless.

Considering the composition and the chain rule for $g^{\prime }(\zeta )\ne 0\iff \zeta \ne \pm \sqrt{m}$ , e.g., with respect to $w$ providing

$$w(g(\zeta ))=\hat{w}(\zeta ),$$

(24)

$$w^{\prime }(z)=\frac{{\hat{w}}^{\prime }(g^{-1}(z))}{g^{\prime }(g^{-1}(z))},$$

(25)

the problem is ultimately formulated in terms of the $\zeta$ -plane, where the potentials ${\hat{w}}^{\mathrm{i}},{\hat{w}}^o\in \mathcal{H}(\mathbb{C}\backslash \{0\})$ and thus $\hat{w}\in \mathcal{H}(\mathbb{C}\backslash (\overline{\mathbb{D}(\sqrt{m})}\cup {\mathbb{S}}^1))$ are sought so that

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{w}}^{\mathrm{o}}+\overline{{\hat{w}}^{\mathrm{o}}}={\hat{w}}^{\mathrm{i}}+\overline{{\hat{w}}^{\mathrm{i}}},}\\ \multicolumn{1}{c}{{\kappa }^{\mathrm{o}}[\overline{{\hat{w}}^{\mathrm{o}}}-{\hat{w}}^{\mathrm{o}}]={\kappa }^{\mathrm{i}}[\overline{{\hat{w}}^{\mathrm{i}}}-{\hat{w}}^{\mathrm{i}}],}\end{array}$$

(26)

on ${\mathbb{S}}^1$ and

$$\frac{{\hat{w}}^{\prime }(\zeta )}{g^{\prime }(\zeta )}\to -\overline{E^{\infty }}=\mathrm{const},|\zeta |\to \infty .$$

(27)

The solution to this problem is finally obtained by expanding the functions ${\hat{w}}^{\mathrm{o}}(\zeta )$ and ${\hat{w}}^{\mathrm{i}}(\zeta )$ into their corresponding Laurent-series ²

$${\hat{w}}^{\mathrm{o}}(\zeta )=a_1\zeta +\sum _{k=1}^{\infty }{a}_{-k}{\zeta }^{-k},$$

(28)

$${\hat{w}}^{\mathrm{i}}(\zeta )=b_{-1}{\zeta }^{-1}+\sum _{k=1}^{\infty }{b}_k{\zeta }^k,$$

(29)

at $\zeta =0$ with $a_l,b_l\in \mathbb{C}$ , where undesired coefficients are set to zero. In order to determine the coefficients $a_k,b_k$ , equations (26) are multiplied by $1/(\sigma -\zeta )$ with $\sigma \in {\mathbb{S}}^1$ and Cauchy integrated with respect to a point $\zeta \in \mathbb{C}\backslash \overline{\mathbb{D}(1)}$ , yielding, e.g.,

$$\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{{\hat{w}}^{\mathrm{o}}(\sigma )}{\sigma -\zeta }\mathrm{d}\sigma +\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{\overline{{\hat{w}}^{\mathrm{o}}(\sigma )}}{\sigma -\zeta }\mathrm{d}\sigma =\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{{\hat{w}}^{\mathrm{i}}(\sigma )}{\sigma -\zeta }\mathrm{d}\sigma +\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{\overline{{\hat{w}}^{\mathrm{i}}(\sigma )}}{\sigma -\zeta }\mathrm{d}\sigma .$$

(30)

The integrals in equation (30) evaluate to

$$\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{{\hat{w}}^{\mathrm{o}}(\sigma )}{\sigma -\zeta }\mathrm{d}\sigma =-\sum _{k=1}^{\infty }{a}_{-k}{\zeta }_{-k},\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{\overline{{\hat{w}}^{\mathrm{o}}(\sigma )}}{\sigma -\zeta }\mathrm{d}\sigma =-\overline{a_1}{\zeta }^{-1},$$

(31)

$$\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{{\hat{w}}^{\mathrm{i}}(\sigma )}{\sigma -\zeta }\mathrm{d}\sigma =-b_{-1}{\zeta }^{-1},\frac{1}{2\pi \mathrm{i}}\underset{{\mathbb{S}}^1}{\int }\frac{\overline{{\hat{w}}^{\mathrm{i}}(\sigma )}}{\sigma -\zeta }\mathrm{d}\sigma =-\sum _{k=1}^{\infty }\overline{b_k}{\zeta }^{-k},$$

(32)

where the Cauchy integral formulae [66] have been exploited. Inserting equations (31) and (32) into equation (30), comparing coefficients and proceeding analogously with the second of equation (26) leads to the equations

$$a_{-1}+\overline{a_1}=b_{-1}+\overline{b_1},$$

(33)

$$a_{-k}=\overline{b_k},\mathrm{for}k>1,$$

(34)

$${\kappa }^{\mathrm{o}}(a_{-1}-\overline{a_1})={\kappa }^{\mathrm{i}}(b_{-1}-\overline{b_1}),$$

(35)

$${\kappa }^{\mathrm{o}}{a}_{-k}=-{\kappa }^{\mathrm{i}}\overline{b_k},\mathrm{for}k>1.$$

(36)

Equations (34) and (36) are equivalent to $a_k,b_k=0$ for $k>1$ , whereas equations (33) and (35) represent two equations for four unknowns $a_{-1},a_1,b_1$ , and $b_{-1}$ . While plugging equation (28) into equation (27) provides the coefficient $a_1=-R{\overline{E}}^{\infty }$ , the final remaining equation is obtained by requiring the complex potential inside the ellipse to be infinitely differentiable for $z\in [-2R\sqrt{m},2R\sqrt{m}]$ . As shown in Appendix 4, since

$${\hat{w}}^{\mathrm{i}}\in \mathcal{H}(\mathbb{D}(1)\backslash \overline{\mathbb{D}(\sqrt{m})})\iff w^{\mathrm{i}}\in \mathcal{H}({\mathrm{\Omega }}_{\mathbb{C}}^{\mathrm{i}}\backslash [-2R\sqrt{m},2R\sqrt{m}]),$$

(37)

this is equivalent to the continuity of $w^i$ at the slit, cf. [33, 27]:

$$\underset{z^{+}\to z}{lim}{w}^{\mathrm{i}}(z^{+})=\underset{z^{-}\to z}{lim}{w}^{\mathrm{i}}(z^{-}),z\in [-2R\sqrt{m},2R\sqrt{m}],\Im (z^{+})>0,\Im (z^{-})<0$$

(38)

$$\iff \underset{\zeta \to \sqrt{m}{e}^{i\phi }}{lim}{\hat{w}}^{\mathrm{i}}(\zeta )=\underset{\zeta \to \sqrt{m}{e}^{-i\phi }}{lim}{\hat{w}}^{\mathrm{i}}(\zeta ),\zeta \in D(g),\phi \in (0,2\pi ).$$

(39)

Upon inserting equation (29), the condition $m{b}_1=b_{-1}$ emerges. Taken together, the remaining coefficients are identified, resulting in the overall solution

$$\hat{w}(\zeta )=\{\begin{array}{c}-C_{\mathrm{i}}R(\zeta +m{\zeta }^{-1}),\zeta \in \mathbb{D}(1)\backslash \overline{\mathbb{D}(\sqrt{m})},\hfill \\ \multicolumn{1}{c}{R(-\overline{E^{\infty }}\zeta +C_{\mathrm{o}}{\zeta }^{-1}),\zeta \in \mathbb{C}\backslash \overline{\mathbb{D}(1)},}\end{array}$$

(40)

with the constants $C_{\mathrm{i}},C_{\mathrm{o}}\in \mathbb{C}$ given by

$$C_{\mathrm{i}}=\frac{2{\kappa }^{\mathrm{o}}[({\kappa }^{\mathrm{o}}+{\kappa }^{\mathrm{i}})\overline{E^{\infty }}-m({\kappa }^{\mathrm{o}}-{\kappa }^{\mathrm{i}})E^{\infty }]}{[{\kappa }^{\mathrm{o}}(1+m)+{\kappa }^{\mathrm{i}}(1-m)][{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)]},$$

(41)

$$C_{\mathrm{o}}=\frac{[({\kappa }^{\mathrm{i}}{)}^2-{({\kappa }^{\mathrm{o}})}^2][1-m^2]E^{\infty }-4m{\kappa }^o{\kappa }^{\mathrm{i}}\overline{E^{\infty }}}{[{\kappa }^{\mathrm{o}}(1+m)+{\kappa }^{\mathrm{i}}(1-m)][{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)]}.$$

(42)

Ultimately, in consideration of equations (13), (22), (24), and (25), the components of the electric field read

$$E_1(x_1,x_2)-\mathrm{i}{E}_2(x_1,x_2)=\{\begin{array}{c}{C}_{\mathrm{i}},\mathbf{x}\in {\mathrm{\Omega }}^{\mathrm{i}},\hfill \\ \multicolumn{1}{c}{\frac{\overline{E^{\infty }}{(g^{-1}(x_1+\mathrm{i}{x}_2))}^2+C_{\mathrm{o}}}{{(g^{-1}(x_1+\mathrm{i}{x}_2))}^2-m},\mathbf{x}\in {\mathrm{\Omega }}^{\mathrm{o}},}\end{array}$$

(43)

where the solution has been continuously extended on the slit $[-2R\sqrt{m},2R\sqrt{m}]\times \{0\}$ for which $g^{-1}(x_1+\mathrm{i}{x}_2)$ is not defined. The electric displacement field $\mathbf{D}$ is obtained by inserting equation (43) into equation (2).

In order to reinforce the importance of accounting for the correct inverse function, Figure 3(a) shows the allocation of $g^{-1}$ to the two solutions ${\zeta }_1$ and ${\zeta }_2$ of equation (21), whereupon dark gray coloring represents $z\in \mathbb{C}$ for which the inverse equals ${\zeta }_1$ , whereas light gray color corresponds to an equivalence with ${\zeta }_2$ . As can be seen, the right half-plane plus the positive part of the imaginary axis belongs to ${\zeta }_1$ , while the negative part of the imaginary axis plus left half-plane appertain to ${\zeta }_2$ . The errors emanating from the incorrect employment of ${\zeta }_1$ instead of $g^{-1}$ in the shape of equation (22) are visualized in Figure 3(b), where vectors of the electric field are plotted for ratios $b/a=0.5$ and ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}=1000$ , the latter being typical for ferroelectric ceramics embedding cavities filled with air or vacuum. While the constant field inside the ellipse ${\mathbf{E}}^i$ is indifferent with respect to the applied inverse function, unphysical directions and magnitudes of the electric field prevail on the left half-plane. Moreover, the aforementioned split of the imaginary axis becomes obvious with erroneous vectors on its negative part. In contrast, application of $g^{-1}$ in equation (43) yields the correct electric field, which is illustrated in Figure 4. As can be inferred from Figure 4(a), the field inside the ellipse is homogeneous, aligned with the direction of the remote load on account of the problem’s symmetry, and its magnitude is significantly larger than that of ${\mathbf{E}}^{\infty }$ . In addition, the field lines circumvent the cavity due to the much smaller permittivity inside and the comparatively large $b/a$ ratio. As a further consequence, due to the continuity of the tangential and normal components of the electric field and electric displacement, respectively, a field concentration is observed at the vertices of the interface, while the co-vertices show a diminution. A qualitatively similar but seemingly rotated picture arises in Figure 4(b) where an oblique load is prescribed. However, although ${\mathbf{E}}^{\mathrm{i}}$ appears to be collinear with ${\mathbf{E}}^1$ , their angles in fact do not coincide.

Figure 3.

Illustration of the inverse function $g^{-1}$ : (a) allocations of $g^{-1}(z)={\zeta }_1(z)$ (dark gray) and $g^{-1}(z)={\zeta }_2(z)$ (light gray) and (b) vectors of the electric field $\mathbf{E}$ using ${\zeta }_1(z)$ of equation (23) in equation (43) for a two-domain dielectric with $b/a=0.5$ and ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}=1000$ subjected to the remote electric load ${\mathbf{E}}^{\infty }$ inclined with respect to the $x_1$ -axis with an angle of ${45}^{°}$ .

Figure 4.

Unit vectors and magnitude of the electric field $\mathbf{E}$ as well as field lines for a two-domain dielectric with $b/a=0.5$ and ${\kappa }^{°}/{\kappa }^{\mathrm{i}}=1000$ subjected to the remote electric load ${\mathbf{E}}^{\infty }$ inclined with respect to the $x_1$ -axis with angles of (a) ${90}^{°}$ and (b) ${45}^{°}$ .

By means of the solution equation (43), several cases relevant for the following investigations may be modeled, bearing in mind an ellipse corresponds to the geometry of a mechanically opened crack in mode I:

• ellipse of finite permittivity:   ${\kappa }^{\mathrm{i}}\ge {\kappa }_0$ and $m\in [0,1)$ ,

• impermeable ellipse:   ${\kappa }^{\mathrm{i}}=0$ and $m\in [0,1)$ ,

• crack of finite permittivity:   ${\kappa }^{\mathrm{i}}\ge {\kappa }_0$ and $m=1$ ,

• impermeable crack:   ${\kappa }^{\mathrm{i}}=0$ and $m\to 1$

It is noteworthy that, in order to obtain the solution of an impermeable crack from equation (43), ${\kappa }^{\mathrm{i}}=0$ has to be inserted first, followed by the limiting process of $m\to 1$ [26]. Reversing this order, i.e., letting first $m\to 1$ and then ${\kappa }^{\mathrm{i}}\to 0$ , results in a homogeneous electric field ${\mathbf{E}}^{\mathrm{o}}$ throughout the body, undisturbed by the presence of the crack.

2.2. Crack boundary conditions

Considering boundary value problems involving a crack devoid of cohesive zones from the solely mechanical viewpoint, its faces constitute Neumann boundary conditions and are usually assumed traction-free. Accounting for electrostatic facets, however, goes hand in hand with additional electric boundary conditions, which turn out less unambiguous than their mechanical counterparts. In particular, upon loading, a crack medium of permittivity ${\kappa }^{\mathrm{C}}$ , oftentimes air or vacuum, is located inside the distorted crack. Within the straightforward generalization of zero stress boundary conditions, the impermeable crack model assumes ${\kappa }^{\mathrm{C}}=0$ , resulting in a vanishing surface charge density $\omega$ on the crack faces [23]. As a consequence, electric field lines completely circumvent the crack. In contrast, it is entirely penetrated by the electric field in case of the permeable crack model, where zero voltage across two opposite points on the crack faces follows from either ${\kappa }^{\mathrm{C}}\to \infty$ or a closed crack with ${\kappa }^C>0$ [26]. In reality, these two models represent extreme cases of a crack medium of finite permittivity ${\kappa }^{\mathrm{C}}\ge {\kappa }_0$ , obeying Maxwell’s equations (3) in the interior as well as interface conditions equation (4) at the crack faces.

2.2.1. Capacitor analogy in the undeformed configuration

Accounting for the crack’s finite permittivity as well as opening due to loading, the semi-permeable model developed by Hao and Shen [37] considers pairwise opposite points on the crack faces as one-dimensional plate capacitors. Considering, for example, a crack of length $2a$ lying in the $x_1$ - $x_2$ plane of a Cartesian coordinate system, in the undeformed configuration dividing the body into upper $(+)$ and lower $(-)$ parts ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$ , respectively, with the crack faces located at $S^{\mathrm{c}}=\{\mathbf{x}\in {\mathbb{R}}^2|x_1\in (-a,a),x_2=0\}$ , the vertical electric field and displacement components on $S^{\mathrm{c}}$ according to the capacitor analogy are

$$E_2^{\mathrm{C}}(\mathbf{x})=-\frac{\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{+}\to \mathbf{x}}{{\mathbf{x}}^{+}\in {\mathrm{\Omega }}^{+}}}{lim}\alpha ({\mathbf{x}}^{+})-\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{-}\to \mathbf{x}}{{\mathbf{x}}^{-}\in {\mathrm{\Omega }}^{-}}}{lim}\alpha ({\mathbf{x}}^{-})}{\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{+}\to \mathbf{x}}{{\mathbf{x}}^{+}\in {\mathrm{\Omega }}^{+}}}{lim}{u}_2({\mathbf{x}}^{+})-\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{-}\to \mathbf{x}}{{\mathbf{x}}^{-}\in {\mathrm{\Omega }}^{-}}}{lim}{u}_2({\mathbf{x}}^{-})},\mathbf{x}\in S^{\mathrm{c}},$$

(44)

$$D_2^{\mathrm{C}}(\mathbf{x})={\kappa }^{\mathrm{C}}{E}_2^{\mathrm{C}}(\mathbf{x}),\mathbf{x}\in S^{\mathrm{c}},$$

(45)

where $u_2$ is the vertical displacement. This approach basically works for a piezoelectric crack problem, where both $\alpha$ and $u_2$ on the crack faces are obtained from the solution of an undeformed slit. The electric displacement $D_2^{\mathrm{C}}$ is then used as surface charge densities ${\omega }^{+}$ and ${\omega }^{-}$ on the crack faces, i.e., we have

$$\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{+}\to \mathbf{x}}{{\mathbf{x}}^{+}\in {\mathrm{\Omega }}^{+}}}{lim}{D}_2({\mathbf{x}}^{+})=\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{-}\to \mathbf{x}}{{\mathbf{x}}^{-}\in {\mathrm{\Omega }}^{-}}}{lim}{D}_2({\mathbf{x}}^{-})=-{\omega }^{+}(\mathbf{x})={\omega }^{-}(\mathbf{x})=D_2^{\mathrm{C}}(\mathbf{x}),\mathbf{x}\in S^{\mathrm{c}}.$$

(46)

Since neither $\alpha (\mathbf{x})$ nor $u_2(\mathbf{x})$ is known a priori, but rather emerge from the solution of the respective boundary value problem, equations (44)–(46) provide nonlinear electric Neumann boundary conditions, which may require an iterative solution.

Complementing the surface charges of equation (46) and building on the work of Kemmer and Balke [39], Landis and McMeeking [40] introduced Coulombic tractions $t_2^{+}$ and $t_2^{-}$ acting on the crack faces. To this end, an electrostatic stress ${\sigma }_{22}^{\mathrm{C}}$ is formally introduced, in case of a linear-isotropic medium reading

$${\sigma }_{22}^{\mathrm{C}}(\mathbf{x})=\frac{1}{2}{E}_2^{\mathrm{C}}(\mathbf{x})D_2^{\mathrm{C}}(\mathbf{x}),\mathbf{x}\in S^{\mathrm{c}}.$$

(47)

It follows that

$$\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{+}\to \mathbf{x}}{{\mathbf{x}}^{+}\in {\mathrm{\Omega }}^{+}}}{lim}{\sigma }_{22}({\mathbf{x}}^{+})=\underset{\genfrac{}{}{0ex}{}{{\mathbf{x}}^{-}\to \mathbf{x}}{{\mathbf{x}}^{-}\in {\mathrm{\Omega }}^{-}}}{lim}{\sigma }_{22}({\mathbf{x}}^{-})=-t_2^{+}(\mathbf{x})=t_2^{-}(\mathbf{x})={\sigma }_{22}^{\mathrm{C}}(\mathbf{x}),\mathbf{x}\in S^{\mathrm{c}}.$$

(48)

Obviously, the crack’s curvature as well as $x_1$ - and $x_3$ -components of the electric field and displacement are not incorporated in equations (46) and (48).

2.2.2. Capacitor analogy in the deformed configuration

Since in a non-coupled dielectric problem the solution of $\alpha$ is only available on the opened crack faces, an ellipse is used in this work to model deformed cracks in a two-dimensional setting. The capacitor analogy, originally working solely with quantities in the undeformed configuration, is thus modified in the following. Let the ${\mathrm{\Gamma }}^{+}$ and ${\mathrm{\Gamma }}^{-}$ denote the opened upper and lower crack faces so that ${\mathrm{\Gamma }}^{+}\cup {\mathrm{\Gamma }}^{-}=\mathrm{\Gamma }\backslash \{\pm a{\mathbf{e}}_1\}$ , then the vertical component of the electric field and electric displacement inside the opened crack are determined as

$$E_2^{\mathrm{C}}(\mathbf{x})=-\frac{{\alpha }^{\mathrm{o}}(x_1,b\sqrt{1-x_1^2/a^2})-{\alpha }^{\mathrm{o}}(x_1,-b\sqrt{1-x_1^2/a^2})}{2b\sqrt{1-x_1^2/a^2}},\mathbf{x}\in \overline{{\mathrm{\Omega }}^i}\backslash \{\pm a{\mathbf{e}}_1\},$$

(49)

$$D_2^{\mathrm{C}}(\mathbf{x})={\kappa }^{\mathrm{C}}{E}_2^{\mathrm{C}}(\mathbf{x}),\mathbf{x}\in \overline{{\mathrm{\Omega }}^{\mathrm{i}}}\backslash \{\pm a{\mathbf{e}}_1\}.$$

(50)

Defining the electrostatic stress in analogy to equation (47), the surface charges and tractions ${\omega }^{\mathrm{C}}$ and ${\mathbf{t}}^{\mathrm{C}}=t_2^{\mathrm{C}}{\mathbf{e}}_2$ , respectively, on the crack faces read

$${\omega }^{\mathrm{C}}=\{\begin{array}{c}{D}_2^{\mathrm{C}}\mathrm{on}{\mathrm{\Gamma }}^{+},\hfill \\ \multicolumn{1}{c}{-D_2^C\mathrm{on}{\mathrm{\Gamma }}^{-},}\end{array}{\mathbf{t}}^C=\{\begin{array}{c}-{\sigma }_{22}^{\mathrm{C}}{\mathbf{e}}_2\mathrm{on}{\mathrm{\Gamma }}^{+},\hfill \\ \multicolumn{1}{c}{{\sigma }_{22}^{\mathrm{C}}{\mathbf{e}}_2\mathrm{on}{\mathrm{\Gamma }}^{-}.}\end{array}$$

(51)

As shown in Appendix 15, in the case of a Griffith crack of length $2a$ in a dielectric under mode I/IV loading, the relative difference of $D_2^{\mathrm{C}}$ calculated according to equations (45) and (50) is less than $1\%$ for relevant crack openings, so that the results to follow are directly comparable to literature where equation (44) is used. Although the curvature of the crack faces apparently does not play a role at this point, the question remains whether the simplifications of the capacitor analogy model themselves are permissible, in particular with regard to inclined electric fields and charges as well as tractions on the crack faces and their implications for crack tip loading. These issues will be illuminated in Section 3.

2.2.3. Electrodynamic action of force

While the second of equations (51) is only strictly valid for infinite straight conducting capacitor plates enclosing a linear isotropic dielectric, it may be put into the more general context of electrodynamic action of force. This phenomenon represents an important aspect when dealing with electromechanical problems since, aside from energy, electromagnetic fields also carry momentum, which has to be accounted for in mechanical balance equations. Unfortunately, the formulations of the electromagnetic momentum density and the stress tensor in polarizable and magnetizable matter are not generally agreed upon among scientists. In fact, notable attempts to identify the correct momentum density trace back to Abraham [41] and Minkowski [42] in the early 20th century, who sparked a controversy unsolved up to date [67].

Nevertheless, there is a consensus which kind of electrodynamical coupling quantities enter the balance laws of continuum mechanics [43, 68, 69]. These will be briefly illuminated in the following, where considerations are confined to electrostatic settings with magnetic fields absent. To begin with, the local balance of linear momentum in static scenarios, neglecting gravitational body forces, valid for every point in the continuum body reads

$$\mathrm{div}\mathit{\sigma }+{\mathbf{f}}^{\mathrm{el}}=0,$$

(52)

where $\mathit{\sigma }$ is the Cauchy stress tensor and ${\mathbf{f}}^{\mathrm{el}}$ the electric body force. Furthermore, considering a polar continuum while discarding spin angular momentum and couple stresses, the local balance of angular momentum simplifies to

$$\mathit{\omega }:\mathit{\sigma }+{\mathbf{c}}^{\mathrm{el}}=0,$$

(53)

with $\mathit{\omega }$ as the third-order permutation tensor and ${\mathbf{c}}^{\mathrm{el}}$ as electric body couple. Consequently, when accounting for these forces and couples, the Cauchy stress tensor is, in general, neither solenoidal nor symmetric. At last, Neumann boundary conditions on the body’s surface with ${\mathbf{e}}^{\mathrm{n}}$ as outer normal unit vector are given by

$${\mathbf{e}}^{\mathrm{n}}·\mathit{\sigma }={\mathbf{t}}^{*}+{\mathbf{t}}^{\mathrm{el}}.$$

(54)

In equation (54), ${\mathbf{t}}^{*}$ and ${\mathbf{t}}^{\mathrm{el}}$ denote mechanical and electric surface tractions, respectively. In the special cases of electro- or magnetostatics, all three electric quantities in equations (52)–(54) deduce from the second-order Maxwell stress tensor $\mathbf{T}$ by means of

$${\mathbf{f}}^{\mathrm{el}}=\mathrm{div}\mathbf{T},$$

(55)

$${\mathbf{c}}^{\mathrm{el}}=\mathit{\omega }:\mathbf{T},$$

(56)

$${\mathbf{t}}^{\mathrm{el}}={\mathbf{e}}^{\mathrm{n}}·[[\mathbf{T}]].$$

(57)

Equations (55)–(57), which may be derived from the resultant electrostatic force and torque acting on arbitrary parts of the body and its boundary, are widely known in the literature [43, 67, 70–72].

There are three predominant formulations of the Maxwell stress tensor according to Lorentz, Einstein-Laub, or Minkowski, referred to with superscripts L, EL, or M, which are given in the electrostatic case by Datsyuk and Pavlyniuk [73]

$${\mathbf{T}}^{\mathrm{L}}={\kappa }_0\mathbf{E}\otimes \mathbf{E}-\frac{1}{2}{\kappa }_0(\mathbf{E}·\mathbf{E})\mathbf{I},$$

(58)

$${\mathbf{T}}^{\mathrm{EL}}=\mathbf{D}\otimes \mathbf{E}-\frac{1}{2}{\kappa }_0(\mathbf{E}·\mathbf{E})\mathbf{I},$$

(59)

$${\mathbf{T}}^{\mathrm{M}}=\mathbf{D}\otimes \mathbf{E}-\frac{1}{2}(\mathbf{E}·\mathbf{D})\mathbf{I},$$

(60)

each of which yields unique forces and couples. In equations (58)– (60), $\mathbf{I}$ denotes the twice contravariant second-order unit tensor. All three formulations are commonly used in contemporary literature [28, 38, 57, 58, 74–79]. As was shown in Ricoeur and Kuna [80], the electric surface tractions according to Minkowski, ${\mathbf{t}}^{\mathrm{M}}$ , acting on the faces of an opened straight crack pass into the Coulombic tractions ${\mathbf{t}}^{\mathrm{C}}$ , as given in equations (47) and (48), for ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0$ by neglecting the crack’s curvature as well as electric field components parallel to the crack plane. Therefore, ${\mathbf{T}}^{\mathrm{M}}$ and the associated quantities ${\mathbf{f}}^{\mathrm{M}},{\mathbf{t}}^{\mathrm{M}}$ , and ${\mathbf{c}}^{\mathrm{M}}$ are selected in the following section as a basis for comparison with the capacitor analogy. For statically loaded, linear isotropic dielectrics obeying equations (2) and (3), it becomes evident that

$${\mathbf{f}}^{\mathrm{M}},{\mathbf{c}}^{\mathrm{M}}=0\mathrm{in}{\mathrm{\Omega }}^{\mathrm{i}}\cup {\mathrm{\Omega }}^{\mathrm{o}}.$$

(61)

Accordingly, only tractions ${\mathbf{t}}^{\mathrm{M}}$ on the interface $\mathrm{\Gamma }$ , owing to the jumps in the electric field and electric displacement, must be accounted for in the mechanical considerations to follow.

3. Results

3.1. Electric field inside the ellipse

The components of the electric field inside the ellipse are obtained in explicit form from equations (41) and (43):

$$E_1^{\mathrm{i}}=\Re (C_{\mathrm{i}})=\frac{2{\kappa }^{\mathrm{o}}}{{\kappa }^{\mathrm{o}}(1+m)+{\kappa }^{\mathrm{i}}(1-m)}{E}_1^{\infty },$$

(62)

$$E_2^{\mathrm{i}}=-\Im (C_{\mathrm{i}})=\frac{2{\kappa }^{\mathrm{o}}}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{E}_2^{\infty }.$$

(63)

Equations (62) and (63) reveal that the electric field components $E_1^{\mathrm{i}}$ and $E_2^{\mathrm{i}}$ are proportional to their corresponding counterparts $E_1^{\infty }$ and $E_2^{\infty }$ , so that, e.g., remote electric loads aligned in $x_1$ -direction do not engender a field $E_2^{\mathrm{i}}$ . A further noteworthy peculiarity is that for ${\kappa }^{\mathrm{i}}>0$ and $m\to 1$ , $E_1^{\mathrm{i}}=E_1^{\infty }$ and $E_2^{\mathrm{i}}={\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}{E}_2^{\infty }$ are obtained. As a consequence of tangential continuity of the electric field, a bounded field concentration of $E_2^{\mathrm{o}}(\pm a,0)/E_2^{\infty }\to {\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ occurs at the vertices [28].

In the following, the influence of an inclination of ${\mathbf{E}}^{\infty }$ on the direction of ${\mathbf{E}}^{\mathrm{i}}$ will be examined in order to determine under which conditions the assumption of a vertical field ${\mathbf{E}}^{\mathrm{i}}$ made within the capacitor analogy is justified. While numerical results indicate the assumption’s validity [55], an analytical proof is still pending. To this end, the angles ${\gamma }^1$ and ${\gamma }^{\mathrm{i}}$ of both vectors enclosing the $x_1$ -axis are introduced according to

$$\tan ({\gamma }^{\infty })=\frac{E_2^{\infty }}{E_1^{\infty }},\tan ({\gamma }^{\mathrm{i}})=\frac{E_2^{\mathrm{i}}}{E_1^{\mathrm{i}}}.$$

(64)

Restricting further considerations to $E_1^{\infty }>0$ , i.e., ${\gamma }^{\infty }\in (-\pi /2,\pi /2)$ , and thus $E_1^{\mathrm{i}}>0$ , allows for the convenient transformation of the second of equations (64) avoiding a distinction of cases, yielding the angle ${\gamma }^{\mathrm{i}}:[0,\infty )\times [0,1)\times (-\pi /2,\pi /2)\to [-\pi /2,\pi /2]$ as ³

$${\gamma }^{\mathrm{i}}(\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}},m,{\gamma }^{\infty })=\arctan \left(\frac{E_2^{\mathrm{i}}}{E_1^{\mathrm{i}}}\right)=\arctan (\frac{(1+m)+\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}(1-m)}{(1-m)+\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}(1+m)}\tan ({\gamma }^{\infty })).$$

(65)

By means of equation (65), several special cases are to be investigated: a crack of finite permittivity corresponds to

$${\gamma }^{\mathrm{i}}(\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}},m=1,{\gamma }^{\infty })=\arctan (\frac{{\kappa }^{\mathrm{o}}}{{\kappa }^{\mathrm{i}}}\tan ({\gamma }^{\infty })),$$

(66)

while an impermeable ellipse results in

$${\gamma }^{\mathrm{i}}(\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}=0,m,{\gamma }^{\infty })=\arctan (\frac{1+m}{1-m}\tan ({\gamma }^{\infty })).$$

(67)

Finally, the impermeable crack leads to

$$\begin{array}{c}\multicolumn{1}{c}{\underset{m\to 1}{lim}{\gamma }^{\mathrm{i}}(\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}=0,m,{\gamma }^{\infty })=\underset{m\to 1}{lim}\arctan (\frac{1+m}{1-m}\tan ({\gamma }^{\infty }))}\\ \multicolumn{1}{c}{=\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}\arctan (\frac{{\kappa }^o}{{\kappa }^i}\tan ({\gamma }^{\infty }))=\mathrm{sgn}({\gamma }^{\infty })\frac{\pi }{2}.}\end{array}$$

(68)

Equation (68) represents the corresponding limiting cases of equations (66) and (67), which both end up with a vertical field inside the ellipse following the sign of $E_2^{\infty }$ if ${\gamma }^{\infty }\ne 0$ . The asymptotic nature of these limits is depicted in Figure 5. First, ${\gamma }^{\mathrm{i}}$ according to equation (66) is plotted versus the ratio ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ in Figure 5(a) for several values of ${\gamma }^{\infty }$ . As can be seen, for small ratios ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\le {10}^2$ , a nearly vertical field inside the crack presupposes a nearly vertical load, whereas a significant alignment with the $x_1$ -axis takes place for well-nigh horizontal loads. By contrast, for ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\ge {10}^3$ , even minuscule contributions of $E_2^{\infty }$ to the electric load result in a quasi-vertical field ${\mathbf{E}}^{\mathrm{i}}$ . Furthermore, equation (67) is illustrated in Figure 5(b), where ${\gamma }^{\infty }={\gamma }^{\mathrm{i}}(m=0)$ holds. For increasing $m$ , the angle ${\gamma }^{\mathrm{i}}$ proceeds monotonously up to $\pm \pi /2$ where finally $m=1$ is approached. The parameter range of virtually vertical fields, however, is confined to a narrow interval close to $m=1$ , and thus only prevails for flat ellipses. In conclusion, for ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}<<1$ and $1-m<<1$ , assuming the field inside the ellipse to be vertical poses an excellent approximation with negligible deviation from the true field ${\mathbf{E}}^{\mathrm{i}}$ . Therefore, with regard to dielectrics exhibiting a high permittivity such as barium titanate (BT), the horizontal component of the electric field inside a closed or mechanically opened crack may be disregarded, confirming the numerical findings in Ricoeur et al. [55].

Figure 5.

Angle of the electric field inside the ellipse ${\gamma }^{\mathrm{i}}$ for various angles of the electric load at infinity ${\gamma }^{\infty }$ : (a) ${\gamma }^{\mathrm{i}}$ versus ${\kappa }^{°}/{\kappa }^{\mathrm{i}}$ in logarithmic scale for the case of a sharp crack $(m=1)$ and (b) ${\gamma }^{\mathrm{i}}$ versus ellipse parameter $m$ in the limiting case of ${\kappa }^{\mathrm{i}}/{\kappa }^{°}=0$ with ${\gamma }^{\mathrm{i}}(m=0)={\gamma }^{\infty }$ .

3.2. Approximate versus true surface charges and tractions

In the following, the approximate charges and tractions are understood to be results of the simple, however established, capacitor analogy model as amended according to Section 2.2.2. We define true quantities as those which emanate from the solution of the elliptic cavity outlined in Sections 2.1 and 3.1.

For further analyses of the surface charge density $\omega$ and tractions ${\mathbf{t}}^{\mathrm{M}}$ , a parameterization ${\mathbf{x}}^{\mathrm{e}}:[0,2\pi ]\to {\mathbb{R}}^2$ of the ellipse $\mathrm{\Gamma }$ in terms of the parameter $\vartheta$ reads

$${\mathbf{x}}^{\mathrm{e}}(\vartheta )=R(1+m)\cos (\vartheta ){\mathbf{e}}_1+R(1-m)\sin (\vartheta ){\mathbf{e}}_2,$$

(69)

where the derivative evaluates to

$$({\mathbf{x}}^{\mathrm{e}}{)}^{\prime }(\vartheta )=-R(1+m)\sin (\vartheta ){\mathbf{e}}_1+R(1-m)\cos (\vartheta ){\mathbf{e}}_2.$$

(70)

By means of equation (70), tangential and normal unit vectors ${\mathbf{e}}^t$ and ${\mathbf{e}}^n$ at the ellipse are specified with

$${\tilde{\mathbf{e}}}^{\mathrm{t}}(\vartheta )={\mathbf{e}}^{\mathrm{t}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))=\frac{({\mathbf{x}}^{\mathit{e}}{)}^{\prime }(\vartheta )}{|({\mathbf{x}}^{\mathit{e}}{)}^{\prime }(\vartheta )|}=\frac{-(1+m)\sin (\vartheta ){\mathbf{e}}_1+(1-m)\cos (\vartheta ){\mathbf{e}}_2}{\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}},$$

(71)

$${\tilde{\mathbf{e}}}^{\mathrm{n}}(\vartheta )={\mathbf{e}}^{\mathrm{n}}({\mathbf{x}}^{\mathit{e}}(\vartheta ))=e_2^{\mathrm{t}}{\mathbf{e}}_1-e_1^{\mathrm{t}}{\mathbf{e}}_2=\frac{(1-m)\cos (\vartheta ){\mathbf{e}}_1+(1+m)\sin (\vartheta ){\mathbf{e}}_2}{\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}}.$$

(72)

Accounting for the direction of ${\mathbf{e}}^{\mathrm{n}}$ pointing inward the dielectric body as depicted in Figure 1, the surface charge density $\tilde{\omega }(\vartheta )=\omega ({\mathbf{x}}^{\mathrm{e}}(\vartheta ))$ at the ellipse is determined using equations (2), (4), (62), (63), and (72):

$$\begin{array}{c}\multicolumn{1}{c}{\tilde{\omega }(\vartheta )={\tilde{\mathbf{e}}}^{\mathrm{n}}(\vartheta )·{\mathbf{D}}^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))={\tilde{\mathbf{e}}}^{\mathrm{n}}(\vartheta )·{\mathbf{D}}^{\mathrm{i}}={\kappa }^{\mathrm{i}}{\tilde{\mathbf{e}}}^{\mathrm{n}}(\vartheta )·{\mathbf{E}}^{\mathrm{i}}}\\ \multicolumn{1}{c}{=\frac{2{\kappa }^{\mathrm{o}}{\kappa }^{\mathrm{i}}}{\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}}(\frac{(1-m)\cos (\vartheta )E_1^{\infty }}{{\kappa }^{\mathrm{o}}(1+m)+{\kappa }^{\mathrm{i}}(1-m)}+\frac{(1+m)\sin (\vartheta )E_2^{\infty }}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}).}\end{array}$$

(73)

Finally, the surface tractions ${\tilde{\mathbf{t}}}^{\mathrm{M}}(\vartheta )={\mathbf{t}}^{\mathrm{M}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))$ are obtained by means of equations (2), (42), (43), (57), (60), and (72), whose components read

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{t}}_1^{\mathrm{M}}(\vartheta )=\frac{1}{2\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}}[(1-m)\cos (\vartheta )[{\kappa }^{\mathrm{o}}((E_1^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta {)))}^2-(E_2^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))}\\ \multicolumn{1}{c}{-{\kappa }^{\mathrm{i}}((E_1^{\mathrm{i}}{)}^2-{(E_2^{\mathrm{i}})}^2)]+2(1+m)\sin (\vartheta )[{\kappa }^{\mathrm{o}}{E}_1^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))E_2^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))-{\kappa }^{\mathrm{i}}{E}_1^{\mathrm{i}}{E}_2^{\mathrm{i}}]],}\end{array}$$

(74)

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{t}}_2^{\mathrm{M}}(\vartheta )=\frac{1}{2\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}}[2(1-m)\cos (\vartheta )[{\kappa }^{\mathrm{o}}{E}_1^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))E_2^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))-{\kappa }^{\mathrm{i}}{E}_1^{\mathrm{i}}{E}_2^{\mathrm{i}}]]}\\ \multicolumn{1}{c}{+(1+m)\sin (\vartheta )[{\kappa }^{\mathrm{o}}((E_2^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta {)))}^2-{(E_1^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}}(\vartheta )))}^2)-{\kappa }^{\mathrm{i}}((E_2^{\mathrm{i}}{)}^2-{(E_1^{\mathrm{i}})}^2)]}\end{array}$$

(75)

While the true fields inside the ellipse as well as charges and tractions at the ellipse are given by equations (62), (63), and (73)–(75), calculation of the approximated quantities $E_2^{\mathrm{C}}$ and $D_2^{\mathrm{C}}$ as well as ${\omega }^{\mathrm{C}}$ and $t_2^{\mathrm{C}}$ according to Section 2.2.2 requires the determination of the electrostatic potential $\alpha$ on the ellipse, which follows from equations (40) and (41):

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }^{\mathrm{o}}({\mathbf{x}}^{\mathrm{e}})={\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})=\Re ({\hat{w}}^{\mathrm{i}}(g^{-1}(x_1^{\mathrm{e}}+\mathrm{i}{x}_2^{\mathrm{e}})))}\\ \multicolumn{1}{c}{=-\frac{2{\kappa }^{\mathrm{o}}{E}_1^{\infty }}{{\kappa }^{\mathrm{o}}(1+m)+{\kappa }^{\mathrm{i}}(1-m)}{x}_1^{\mathrm{e}}-\frac{2{\kappa }^{\mathrm{o}}{E}_2^{\infty }}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{x}_2^{\mathrm{e}}.}\end{array}$$

(76)

Evaluating equation (76) at two opposite points of the ellipse symmetric with respect to the $x_1$ -axis renders the corresponding voltage

$$\begin{array}{c}\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})={\alpha }^{\mathrm{i}}(x_1^{\mathrm{e}},x_2^{\mathrm{e}})-{\alpha }^{\mathrm{i}}(x_1^{\mathrm{e}},-x_2^{\mathrm{e}})=-\frac{4{\kappa }^{\mathrm{o}}{E}_2^{\infty }}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{x}_2^{\mathrm{e}}\hfill \end{array}$$

(77)

between both points. Due to the voltage’s linear dependence on $x_2^{\mathrm{e}}$ , the two variables $x_1^{\mathrm{e}}$ and $\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})$ obviously satisfy the ellipse equation

$${\left(\frac{x_1^{\mathrm{e}}}{R(1+m)}\right)}^2+{(\frac{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{4{\kappa }^{\mathrm{o}}{E}_2^{1}R(1-m)}\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}}))}^2=1,$$

(78)

bearing in mind that $a=R(1+m)$ . The different behaviors of ${\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})$ and $\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})$ with respect to $x_1^{\mathrm{e}}$ are illustrated in Figure 6. Equation (77) is used to compute the electric field $E_2^{\mathrm{C}}$ and electric displacement $D_2^{\mathrm{C}}$ according to the capacitor analogy, equation (50), by inserting $x_2^{\mathrm{e}}=b\sqrt{1-{(x_1^{\mathrm{e}})}^2/a^2}$ :

$$E_2^{\mathrm{C}}=-\frac{\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})}{2x_2^{\mathrm{e}}}=\frac{2{\kappa }^{\mathrm{o}}}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{E}_2^{\infty },$$

(79)

$$D_2^{\mathrm{C}}={\kappa }^{\mathrm{i}}{E}_2^{\mathrm{C}}=\frac{2{\kappa }^{\mathrm{o}}{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{E}_2^{\infty }.$$

(80)

Figure 6.

Infinite isotropic dielectric with elliptic cutout subjected to inclined electric load ( ${\gamma }^1=\pi /4$ ): graphs of ${\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})$ and $\mathrm{\Delta }{\alpha }^{\mathrm{i}}({\mathbf{x}}^{\mathrm{e}})$ versus $x_1^{\mathrm{e}}$ .

As stated beforehand, the $x_1$ -components of the electric field and electric displacement, see equation (62), are erroneously neglected in the case of $E_1^{\infty }\ne 0$ . However, as previously pointed out, the component $E_1^{\mathrm{i}}$ may be of negligible magnitude for crack problems of interest. Per contra, equations (63), (79), and (80) reveal that the $x_2$ -components are exact, i.e., $E_2^{\mathrm{C}}=E_2^{\mathrm{i}}$ and $D_2^{\mathrm{C}}=D_2^{\mathrm{i}}$ , cf. [23]. Moreover, via equations (51) and (80), the approximated surface charge density ${\omega }^{\mathrm{C}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))={\tilde{\omega }}^{\mathrm{C}}(\vartheta )$ evaluates to

$$\begin{array}{c}{\tilde{\omega }}^{\mathrm{C}}(\vartheta )=\frac{\sin (\vartheta )}{|\sin (\vartheta )|}{D}_2^{\mathrm{C}}=\frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{2{\kappa }^{\mathrm{o}}{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{E}_2^{\infty },\hfill \end{array}$$

(81)

keeping in mind that $\vartheta \in (0,\pi )$ and $\vartheta \in (\pi ,2\pi )$ represent locations in the positive ( $x_2^{\mathrm{e}}>0$ ) and negative $(x_2^{\mathrm{e}}<0)$ half planes, respectively, corresponding to ${\omega }^{+}$ and ${\omega }^{-}$ . It becomes obvious that $\omega$ and ${\omega }^{\mathrm{C}}$ do not only depend on the quotient ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}$ , but on both permittivities independently. Plugging $\vartheta \in \{\pi /2,3\pi /2\}$ into equations (73) and (81) while setting $E_1^{\infty }=0$ shows that ${\omega }^C$ is exact at the co-vertices $(x_1^{\mathrm{e}}=0)$ constituting the upper and lower bounds of $\omega$ , respectively:

$$\begin{array}{c}\multicolumn{1}{c}{\tilde{\omega }(\vartheta =\frac{\pi }{2})={\tilde{\omega }}^{\mathrm{C}}(\vartheta =\frac{\pi }{2})=-\tilde{\omega }(\vartheta =\frac{3\pi }{2})=-{\tilde{\omega }}^{\mathrm{C}}(\vartheta =\frac{3\pi }{2})}\\ \multicolumn{1}{c}{=\frac{2{\kappa }^{\mathrm{o}}{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}{E}_2^{\infty }.}\end{array}$$

(82)

Beyond that, approaching conditions of a closed permeable crack for $m\to 1$ , the approximation becomes exact on the entire ellipse regardless of the ratio ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\ne 0$ , i.e.,

$$\underset{m\to 1}{lim}\tilde{\omega }(\vartheta )=\underset{m\to 1}{lim}{\tilde{\omega }}^C(\vartheta )=\frac{\sin (\vartheta )}{|\sin (\vartheta )|}{\kappa }^{\mathrm{o}}{E}_2^{\infty },$$

(83)

which is appropriate for a homogeneous field ${\mathbf{E}}^{\mathrm{o}}(\mathbf{x})={\mathbf{E}}^{\infty }$ in ${\mathrm{\Omega }}^{\mathrm{o}}$ . With regard to an ellipse with $m<1$ , however, ${\omega }^{\mathrm{C}}$ does not converge to the true surface charges for impermeable conditions ${\kappa }^i/{\kappa }^o\to 0$ , i.e.,

$$\begin{array}{c}\multicolumn{1}{c}{\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}\tilde{\omega }(\vartheta )=\frac{2{\kappa }^{\mathrm{i}}}{\sqrt{{(1+m)}^2-4m\stackrel{2}{\cos }(\vartheta )}}(\frac{1-m}{1+m}\cos (\vartheta )E_1^{\infty }+\frac{1+m}{1-m}\sin (\vartheta )E_2^{\infty })}\\ \multicolumn{1}{c}{\ne \frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{2{\kappa }^{\mathrm{i}}}{1-m}{E}_2^{\infty }=\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}{\tilde{\omega }}^{\mathrm{C}}(\vartheta )}\end{array}$$

(84)

for arbitrary parameters $m,\vartheta ,{\kappa }^{\mathrm{i}}\ne 0,E_1^{\infty }$ , and $E_2^{\infty }$ . The approximation’s quality is further illuminated in Figure 7 for an inclined load, where the normalized surface charge ⁴ $\tilde{\omega }/|{\tilde{\omega }}^{\mathrm{C}}|$ is plotted versus the ratio ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ for a pronounced ellipse with $m=0.5$ in Figure 7(a) versus the parameter $m$ for a small ratio of ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}=10$ in Figure 7(b) with the singularity at $\vartheta =0,\pi ,2\pi$ removed by continuous extension.

Figure 7.

Infinite isotropic dielectric $({\kappa }^{°})$ with elliptic cavity $\left({\kappa }^{\mathrm{i}}\right)$ subjected to an oblique remote load ${\mathbf{E}}^1$ with ${\gamma }^1=\pi /4$ : true surface charge density $\tilde{\omega }$ at the ellipse normalized with respect to the absolute value of the capacitor analogy ${\tilde{\omega }}^{\mathrm{C}}$ : (a) plotted versus the common logarithm of the ratio ${\kappa }^{°}/{\kappa }^{\mathrm{i}}$ $(m=0.5)$ and (b) plotted versus slenderness parameter $m$ $({\kappa }^{°}/{\kappa }^{\mathrm{i}}=10)$ .

The former brings out that the ratio $\tilde{\omega }/|{\tilde{\omega }}^{\mathrm{C}}|$ is insensitive to the quotient of permittivities, whereas the latter highlights the speed of convergence for $m\to 1$ . Most notably, maximal relative underestimations by ${\tilde{\omega }}^{\mathrm{C}}$ are bounded by roughly $50\%$ . In conclusion, enforcement of the surface charge density ${\tilde{\omega }}^{\mathrm{C}}$ as electric Neumann boundary condition provides a good substitute for the real electric interface conditions for a crack-like slit, i.e., if $1-m<<1$ , independently of the ratio of permittivities ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ and the angle ${\gamma }^{\infty }$ .

The Coulombic traction $t_2^{\mathrm{C}}({\mathbf{x}}^{\mathrm{e}}(\vartheta ))={\tilde{t}}_2^{\mathrm{C}}(\vartheta )$ is obtained by inserting equations (79) and (80) into equations (47) and (51), respectively, reading

$${\tilde{t}}_2^{\mathrm{C}}(\vartheta )=-\frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{1}{2}{E}_2^{\mathrm{C}}{D}_2^{\mathrm{C}}=-\frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{2({\kappa }^{\mathrm{o}}{)}^2{\kappa }^{\mathrm{i}}}{{[{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)]}^2}(E_2^{\infty }{)}^2,$$

(85)

with an allocation of $t_2^{+}$ and $t_2^{-}$ analogous as in equation (81). Analogous to the true and the approximated surface charges, sole prescription of the ratio ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ is insufficient to determine ${\mathbf{t}}^{\mathrm{M}}$ and $t_2^{\mathrm{C}}$ . Considering the limiting case of a slit crack in equations (74), (75), and (85), cf. [81] for $E_3^{\mathrm{C}}=0$ , ends up with

$$\underset{m\to 1}{lim}{\tilde{t}}_2^{\mathrm{C}}(\vartheta )=-\frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{1}{2}\frac{{({\kappa }^{\mathrm{o}})}^2}{{\kappa }^{\mathrm{i}}}(E_2^{\infty }{)}^2,$$

(86)

$$\underset{m\to 1}{lim}{\tilde{t}}_1^{\mathrm{M}}(\vartheta )=0,$$

(87)

$$\underset{m\to 1}{lim}{\tilde{t}}_2^{\mathrm{M}}(\vartheta )=\frac{\sin (\vartheta )}{|\sin (\vartheta )|}\frac{1}{2}{\kappa }^{\mathrm{o}}[(\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}-1){(E_1^{\infty })}^2+(1-\frac{{\kappa }^{\mathrm{o}}}{{\kappa }^{\mathrm{i}}}){(E_2^{\infty })}^2].$$

(88)

First, ${\mathbf{t}}^{\mathrm{M}}$ is constant at the upper and lower crack faces. While the $x_1$ -component of ${\mathbf{t}}^{\mathrm{M}}$ is zero according to equation (87), the $x_2$ -component is, on the one hand, of vanishing magnitude for ${\kappa }^{\mathrm{o}}\to {\kappa }^{\mathrm{i}}$ and, on the other hand, dominated by the contribution of $E_2^{\infty }$ for $|E_1^{\infty }|\approx |E_2^{\infty }|$ and ${\kappa }^{\mathrm{o}}>>{\kappa }^{\mathrm{i}}$ . In particular, for $E_2^{\infty }\ne 0$ and ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0$ , we find for all $\vartheta$ , cf. [82]:

$$\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}({\tilde{t}}_2^{\mathrm{M}}(\vartheta ){|}_{m=1}-{\tilde{t}}_2^{\mathrm{C}}(\vartheta ){|}_{m=1})=0,$$

(89)

$$\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}|{\tilde{t}}_2^{\mathrm{M}}(\vartheta ){|}_{m=1}|=\underset{{\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}\to 0}{lim}|{\tilde{t}}_2^{\mathrm{C}}(\vartheta ){|}_{m=1}|=\infty .$$

(90)

In order to estimate under which circumstances the closing effect of the Minkowski traction $t_2^{\mathrm{M}}$ acting on a crack filled with vacuum is noticeable compared to mechanical loads, its sensitivity towards the dielectric constant ${\kappa }^{\mathrm{o}}$ and the electric load’s angle ${\gamma }^{\infty }$ is examined in Figure 8(a). The load’s intensity is prescribed with $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ , simulating typical upper limits of electric fields applied in experiments [83–86]. A nearly linear behavior of magnitude versus ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ is observed in a log–log plot for ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\ge 10$ , independent of the load’s inclination. Furthermore, relevant magnitudes in the region of several MPa only prevail for large dielectric constants ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\ge {10}^3$ , where the value of unpoled and thus isotropic BT is highlighted for reference. Moreover, according to Figure 8(b), where the ratio of Minkowski to Coulombic traction is plotted, $t_2^{\mathrm{C}}$ poses a poor approximation of $t_2^{\mathrm{M}}$ for small permittivities ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\le 10$ , for which, however, the tractions’ magnitude is negligible. On the contrary, the approximation quickly converges to the exact value, regardless of the electric load’s angle, for ${\kappa }^o/{\kappa }^i\to 1$ . All things considered, the error turns out marginal for ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}\ge {10}^2$ .

Figure 8.

Infinite cracked dielectric $(m=1,{\kappa }^{\mathrm{i}}={\kappa }_0)$ subjected to oblique remote load ${\mathbf{E}}^{\infty }$ with $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ : (a) absolute value of ${\tilde{t}}_2^{\mathrm{M}}$ and (b) Minkowski traction ${\tilde{t}}_2^{\mathrm{M}}$ normalized with respect to the Coulombic traction ${\tilde{t}}_2^{\mathrm{C}}$ versus ratio ${\kappa }^{°}/{\kappa }^{\mathrm{i}}$ for various angles ${\gamma }^{\infty }$ ; permittivity of BT highlighted for reference.

On the other hand, accounting for slender ellipses with $1-m<<1$ , simulating a crack opening due to tensile loads, Figure 9 reveals drastic deviations of the Minkowski from the Coulombic tractions due to curvature when approaching the crack tip, where the former’s magnitude exceeds the latter’s by several orders. This owes to the previously mentioned field concentration $E_2^{\mathrm{o}}/E_2^{\infty }$ near the vertices for $m\to 1$ tending to the factor ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ , which amounts to roughly ${10}^3$ for BT [26]. However, these large errors are constricted to the immediate neighbourhood of the vertices, which raises the question whether or not they are relevant for crack tip loading of opened cracks.

Figure 9.

BT exhibiting an elliptic cavity filled with vacuum $(m\le 1,{\kappa }^{\mathrm{i}}={\kappa }_0)$ subjected to oblique remote load ${\mathbf{E}}^{\infty }$ with $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ and ${\gamma }^{\infty }=\pi /4$ : absolute values of (a) the Minkowski traction ${\tilde{t}}_1^{\mathrm{M}}$ and (b) ${\tilde{t}}_2^{\mathrm{M}}$ , normalized with respect to the Coulombic traction ${\tilde{t}}_2^{\mathrm{C}}$ versus slenderness parameter $m$ .

3.3. Investigation of intensity factors

In order to assess this point, consider a straight crack of length $l=2a$ in the undeformed configuration of an isotropic linear-elastic dielectric body aligned with the $x_1$ -axis of a Cartesian coordinate system with origin at the crack’s center. The mechanical stress ${\sigma }_{22}^{\infty }$ is applied at infinity, see left sketch of Figure 10, and the body deforms according to the displacement field $\mathbf{u}(\mathbf{x})$ , resulting in a distorted crack of elliptic shape, which, in a state of plane strain, yields the following ellipse parameters as functions of the mechanical load [87]:

$$\tilde{a}({\sigma }_{22}^{\infty })=a+\underset{x_2\downarrow 0}{lim}{u}_1(a,x_2)=(1-\frac{1-\nu }{2\mu }{\sigma }_{22}^{\infty })a,$$

(91)

$$\tilde{b}({\sigma }_{22}^{\infty })=\underset{x_2\downarrow 0}{lim}{u}_2(0,x_2)=\frac{1-\nu }{\mu }{\sigma }_{22}^{\infty }a,$$

(92)

or, equivalently,

$$\tilde{R}({\sigma }_{22}^{\infty })=\frac{1}{2}(1+\frac{1-\nu }{2\mu }{\sigma }_{22}^{\infty })a,$$

(93)

$$\tilde{m}({\sigma }_{22}^{\infty })=\frac{1-\frac{3(1-\nu )}{2\mu }{\sigma }_{22}^{\infty }}{1+\frac{1-\nu }{2\mu }{\sigma }_{22}^{\infty }},$$

(94)

Figure 10.

Crack configuration accounting for a mechanically opened crack: a Griffith crack of length $2a$ is first distorted into an ellipse by the remote load ${\sigma }_{22}^{\infty }$ with parameters $\tilde{R}({\sigma }_{22}^{\infty })$ and $\tilde{m}({\sigma }_{22}^{\infty })$ , which are used to determine the Minkowski tractions ${\mathbf{t}}^{\mathrm{M}}$ and surface charges $\omega$ at the ellipse induced by the electric field at infinity ${\mathbf{E}}^{\infty }$ ; finally, the loads at infinity and at the crack faces are imposed at the undeformed configuration.

with $\nu$ and $\mu$ as Poisson’s ratio and the shear modulus, respectively. Let the electric field ${\mathbf{E}}^{\infty }$ be applied in this current configuration, inducing the loads ${\mathbf{t}}^{\mathrm{M}}$ and $\omega$ at the opened crack as depicted in the center of Figure 10. In the last step, consider the same crack in the undeformed state, electromechanically loaded at infinity via ${\sigma }_{22}^{\infty }$ and ${\mathbf{E}}^{\infty }$ , as well as subjected to the previously determined crack face loads ${\mathbf{t}}^{\mathrm{M}}$ and $\omega$ as illustrated in the right sketch, for which the stress and dielectric intensity factors $K_I,K_{\mathrm{II}}$ , and $K_{\mathrm{IV}}$ will be calculated with respect to the right crack tip. It is clear that this approach neglects the influence of ${\mathbf{t}}^{\mathrm{M}}$ on the crack deformation, which, in light of the tractions’ nonuniform distribution, departs in an unknown manner from the elliptic shape caused by sole application of mode I loads at infinity. Notwithstanding, according to the fundamental finding by Williams [88], a sharp crack in a linear-elastic material will always result in a displacement field $\mathbf{u}(\mathbf{x})$ whose series expansion in polar coordinates $(r,\varphi )$ about the crack tip is dominated by a $r^{1/2}$ term for $r\to 0$ , which, in turn, ensures an elliptic opening close to the tip. Therefore, results might suffer quantitatively from not accounting for displacements due to ${\mathbf{t}}^M$ . The qualitative insights, however, possess general validity since the relevant curvature near the tip is modeled accurately.

3.3.1. Electric displacement intensity factor

The SIF and DIF for infinite dielectrics under remote mode I, II, and IV loading read [23, 31, 84]

$$K_{\mathrm{IV}}^{(1)}=\sqrt{\pi a}{D}_2^{\infty },K_I^{(1)}=\sqrt{\pi a}{\sigma }_{22}^{\infty },K_{\mathrm{II}}^{(1)}=\sqrt{\pi a}{\sigma }_{12}^{\infty },$$

(95)

where the superscript $(1)$ refers to the contributions of the remote external loads. The surface charge densities ${\omega }^{\pm }$ contribute to the DIF according to

$$K_{\mathrm{IV}}^{(2)}=\underset{S^c}{\int }({\omega }^{+}{h}_{\mathrm{IV}}^{+}+{\omega }^{-}{h}_{\mathrm{IV}}^{-})\mathrm{d}s=\frac{{\kappa }^{\mathrm{o}}}{K_{\mathrm{IV}}^{(1)}}\underset{S^{\mathrm{c}}}{\int }({\omega }^{+}{D}_l{\alpha }^{(1),+}+{\omega }^{-}{D}_l{\alpha }^{(1),-})\mathrm{d}s,$$

(96)

where + and − refer to limit values of the upper and lower crack faces, respectively, $h_{\mathrm{IV}}^{\pm }$ are the mode IV crack weight functions [89] and the electrostatic potential ${\alpha }^{(1)}$ as well as $K_{\mathrm{IV}}^{(1)}$ are those of the Griffith crack with charge free crack faces, see equations (14) and (40) for ${\kappa }^i=0$ and $m\to 1$ , serving as reference problem [87]. The derivative in equation (96) refers to a coordinate system with the origin at the left crack tip. A transformation $\mathbf{x}({\mathbf{x}}^{*},l)={\mathbf{x}}^{*}-\frac{l}{2}{\mathbf{e}}_1$ is thus required. Accounting for an explicit dependency of the electrostatic potential on the crack length $l$ by writing $\alpha (\mathbf{x},l)$ , the derivative is formed as follows:

$$D_l\alpha (\mathbf{x},l)\frac{d}{\mathrm{dl}}\alpha (\mathbf{x}({\mathbf{x}}^{*},l),l)=\frac{\partial \alpha }{\partial l}(\mathbf{x},l)+\frac{1}{2}{E}_1(\mathbf{x},l).$$

(97)

Parameterizing $S^{\mathrm{c}}$ by means of

$${\mathbf{x}}^{\mathrm{C}}(\vartheta )={\mathbf{x}}^{\mathrm{e}}(\vartheta ){|}_{m=1}=a\cos (\vartheta ){\mathbf{e}}_1,\vartheta \in [0,\pi ],$$

(98)

the derivatives on the crack faces are eventually evaluated to

$$D_l{\alpha }^{(1),\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\sqrt{\pi a}{E}_2^{\infty }{h}_{\mathrm{IV}}^{\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\mp \frac{1}{2}\sqrt{\frac{a+x_1^{\mathrm{C}}(\vartheta )}{a-x_1^{\mathrm{C}}(\vartheta )}}{E}_2^{\infty }=\mp \frac{1}{2}\frac{1+\cos (\vartheta )}{|\sin (\vartheta )|}{E}_2^{\infty }.$$

(99)

The normalized dielectric crack weight functions of equation (99) are plotted in Figure 11. They are unbounded near the right crack tip and tend towards zero when approaching the left crack tip. Consequently, charges close to the crack tip of interest have a much greater impact than those far from it. Furthermore, ${\omega }^{+}({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\tilde{\omega }(\vartheta ),{\omega }^{-}({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\tilde{\omega }(2\pi -\vartheta )$ as well as

$$\mathrm{ds}=|({\mathbf{x}}^C{)}^{\prime }(\vartheta )|\mathrm{d}\vartheta =a|\sin (\vartheta )|\mathrm{d}\vartheta$$

(100)

are specified ⁵ so that the DIF reads

$$\begin{array}{c}\multicolumn{1}{c}{K_{\mathrm{IV}}^{(2)}=-\frac{1}{2}\sqrt{\frac{a}{\pi }}{\int }_0^{\pi }[\tilde{\omega }(\vartheta )-\tilde{\omega }(2\pi -\vartheta )](1+\cos (\vartheta ))\mathrm{ds}}\\ \multicolumn{1}{c}{=-\frac{1}{2}\sqrt{\frac{a}{\pi }}({\int }_0^{\pi }\tilde{\omega }(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta -{\int }_{\pi }^{2\pi }\tilde{\omega }(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta )..}\end{array}$$

(101)

Figure 11.

Normalized dielectric crack weight function $2\sqrt{\pi a}{E}_2^{\infty }{h}_{\mathrm{IV}}^{\pm }$ plotted versus (a) normalized coordinate $x_1^{\mathrm{C}}/a$ and (b) parameter $\vartheta$ .

The integrals of equation (101) are calculated analytically, finally resulting in

$$\begin{array}{c}\multicolumn{1}{c}{K_{\mathrm{IV}}^{(2)}=-\frac{2{\kappa }^{\mathrm{o}}{\kappa }^{\mathrm{i}}(1+m)E_2^{\infty }}{{\kappa }^{\mathrm{o}}(1-m)+{\kappa }^{\mathrm{i}}(1+m)}\sqrt{\frac{a}{\pi m}}\arcsin \left(\frac{2\sqrt{m}}{1+m}\right)}\\ \multicolumn{1}{c}{=-\sqrt{\pi a}{D}_2^{\mathrm{C}}\frac{1+m}{\pi \sqrt{m}}\arcsin \left(\frac{2\sqrt{m}}{1+m}\right).}\end{array}$$

(102)

Equation (102) reveals that $K_{\mathrm{IV}}^{(2)}$ does not depend on $E_1^{\infty }$ . In addition, considering the intensity factor $K_{\mathrm{IV}}^{\mathrm{C}}$ due to ${\omega }^{\mathrm{C}}$ of the capacitor analogy, which is simply given by $K_{\mathrm{IV}}^{\mathrm{C}}=-\sqrt{\pi a}{D}_2^{\mathrm{C}}$ , the ratio of both factors reads

$$\frac{K_{\mathrm{IV}}^{(2)}}{K_{\mathrm{IV}}^C}=\frac{1+m}{\pi \sqrt{m}}\arcsin \left(\frac{2\sqrt{m}}{1+m}\right).$$

(103)

In light of equation (103), the true intensity factor $K_{\mathrm{IV}}^{(2)}$ consists of the approximate factor $K_{\mathrm{IV}}^{\mathrm{C}}$ times a geometric correction term accounting for the elliptic shape of the crack faces and thus only depending on $m$ . Upon combination of equations (95) and (102), the total DIF is obtained as

$$K_{\mathrm{IV}}=K_{\mathrm{IV}}^{(1)}+K_{\mathrm{IV}}^{(2)}=\sqrt{\pi a}{D}_2^{\infty }[1-\frac{\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}2(1+m)}{(1-m)+\frac{{\kappa }^{\mathrm{i}}}{{\kappa }^{\mathrm{o}}}(1+m)}\frac{1}{\pi \sqrt{m}}\arcsin \left(\frac{2\sqrt{m}}{1+m}\right)].$$

(104)

Inserting the slenderness parameter $m=\tilde{m}({\sigma }_{22}^{\infty })$ from equation (94) into equations (103) and (104) yields the functions $K_{\mathrm{IV}}({\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}},{\sigma }_{22}^{\infty })$ and $K_{\mathrm{IV}}^{(2)}/K_{\mathrm{IV}}^{\mathrm{C}}({\sigma }_{22}^{\infty })$ , which are illustrated in Figure 12 employing mechanical properties of BT ⁶ . With regard to the total DIF depicted in Figure 12(a), the three distinct crack conditions introduced in Section 2 are to be distinguished:

• permeable crack:   $K_{\mathrm{IV}}/(\sqrt{\pi a}{D}_2^{\infty })=0$ for ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}>0$ and ${\sigma }_{22}^{\infty }=0$ ,

• semi-permeable crack:   $K_{\mathrm{IV}}/(\sqrt{\pi a}{D}_2^{\infty })\in (0,1)$ for ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}>0$ and ${\sigma }_{22}^{\infty }>0$ ,

• impermeable crack:   $K_{\mathrm{IV}}/(\sqrt{\pi a}{D}_2^{\infty })=1$ for ${\kappa }^{\mathrm{i}}/{\kappa }^o=0$ and ${\sigma }_{22}^{\infty }>0$ .

Figure 12.

Investigation of DIFs: (a) total DIF $K_{\mathrm{IV}}=K_{\mathrm{IV}}^{(1)}+K_{\mathrm{IV}}^{(2)}$ normalized with respect to $K_{\mathrm{IV}}^{(1)}$ as function of stress load ${\sigma }_{22}^1$ and the common logarithm of the ratio ${\kappa }^{°}/{\kappa }^i$ and (b) quotient $K_{\mathrm{IV}}^{(2)}$ due to true crack face charges $\omega$ and $K_{\mathrm{IV}}^C$ due to ${\omega }^C$ according to the capacitor analogy.

To begin with, $K_{\mathrm{IV}}$ is zero for a closed permeable crack and rises for increasing ${\sigma }_{22}^{\infty }$ and ${\kappa }^{\mathrm{o}}/{\kappa }^{\mathrm{i}}$ , reflecting the semi-permeable nature of an opened crack. Ultimately, for ${\kappa }^{\mathrm{i}}/{\kappa }^{\mathrm{o}}=0$ , impermeable conditions are recovered. The approximation’s quality, on the other hand, can be extracted from Figure 12(b), whereupon for ${\sigma }_{22}^{\infty }>0$ $K_{\mathrm{IV}}^{\mathrm{C}}$ slightly overestimates $K_{\mathrm{IV}}^{(2)}$ , as was to be expected in light of equation (82). However, for relevant stresses of up to $50$ MPa, which are in accordance with experiments conducted in the literature [23,85,90], the deviation amounts to less than one per mille, rendering the differences between $K_{\mathrm{IV}}^{(2)}$ and $K_{\mathrm{IV}}^{\mathrm{C}}$ negligible, which confirms the numerical findings in Wippler et al. [54].

3.3.2. Stress intensity factors

Proceeding with the mechanical SIF, the contributions of the traction fields ${\mathbf{t}}^{\pm }$ along the crack faces $S^{\mathrm{c}}$ evaluate to Kuna [23]

$$K_I^{(2)}=\underset{S^c}{\int }({\mathbf{t}}^{+}·{\mathbf{h}}_I^{+}+{\mathbf{t}}^{-}·{\mathbf{h}}_I^{-})\mathrm{d}s=\frac{\mu }{K_I^{(1)}(1-\nu )}\underset{S^{\mathrm{c}}}{\int }({\mathbf{t}}^{+}·D_l{\mathbf{u}}^{(1a),+}+{\mathbf{t}}^{-}·D_l{\mathbf{u}}^{(1a),-})\mathrm{d}s,$$

(105)

$$K_{\mathrm{II}}^{(2)}=\underset{S^{\mathrm{c}}}{\int }({\mathbf{t}}^{+}·{\mathbf{h}}_{\mathrm{II}}^{+}+{\mathbf{t}}^{-}·{\mathbf{h}}_{\mathrm{II}}^{-})\mathrm{d}s=\frac{\mu }{K_{\mathrm{II}}^{(1)}(1-\nu )}\underset{S^{\mathrm{c}}}{\int }({\mathbf{t}}^{+}·D_l{\mathbf{u}}^{(1b),+}+{\mathbf{t}}^{-}·D_l{\mathbf{u}}^{(1b),-})\mathrm{d}s,$$

(106)

with ${\mathbf{h}}_I^{\pm }$ and ${\mathbf{h}}_{\mathrm{II}}^{\pm }$ representing the mode I and II, respectively, crack weight functions. The displacement derivatives for reference problems under pure mode I or II loading at infinity are given componentwise for the parameterization ${\mathbf{x}}^{\mathrm{C}}(\vartheta )$ in terms of

$$D_l{u}_1^{(1a),\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\frac{1-\nu }{4\mu }{\sigma }_{22}^{\infty },D_l{u}_2^{(1a),\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\pm \frac{1+\cos (\vartheta )}{|\sin (\vartheta )|}\frac{1-\nu }{2\mu }{\sigma }_{22}^{\infty },$$

(107)

$$D_l{u}_1^{(1b),\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=\pm \frac{1+\cos (\vartheta )}{|\sin (\vartheta )|}\frac{1-\nu }{2\mu }{\sigma }_{12}^{\infty },D_l{u}_2^{(1b),\pm }({\mathbf{x}}^{\mathrm{C}}(\vartheta ))=-\frac{1-\nu }{2\mu }{\sigma }_{12}^{\infty }.$$

(108)

Since the derivatives in the first of equation (107) and the second of equation (108) turn out constant on $S^{\mathrm{c}}$ and are identical on both crack faces, it becomes evident that the traction components $t_1^{\pm }$ and $t_2^{\pm }$ do not contribute to $K_I^{(2)}$ and $K_{\mathrm{II}}^{(2)}$ , respectively, in light of static equilibrium requiring

$$\underset{S^{\mathrm{c}}}{\int }({\mathbf{t}}^{+}+{\mathbf{t}}^{-})\mathrm{ds}=0.$$

(109)

Finally, collecting equation (95) and equations (105)-(109) as well as inserting ${\mathbf{t}}^{+}({\mathbf{x}}^{\mathrm{C}}(\vartheta ))={\tilde{\mathbf{t}}}^{\mathrm{M}}(\vartheta )$ and ${\mathbf{t}}^{-}({\mathbf{x}}^{\mathrm{C}}(\vartheta ))={\tilde{\mathbf{t}}}^{\mathrm{M}}(2\pi -\vartheta )$ ends up with the formulae

$$K_I^{(2)}=\frac{1}{2}\sqrt{\frac{a}{\pi }}(\underset{0}{\overset{\pi }{\int }}{\tilde{t}}_2^{\mathrm{M}}(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta -\underset{\pi }{\overset{2\pi }{\int }}{\tilde{t}}_2^{\mathrm{M}}(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta ),$$

(110)

$$K_{\mathrm{II}}^{(2)}=\frac{1}{2}\sqrt{\frac{a}{\pi }}(\underset{0}{\overset{\pi }{\int }}{\tilde{t}}_1^{\mathrm{M}}(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta -\underset{\pi }{\overset{2\pi }{\int }}{\tilde{t}}_1^{\mathrm{M}}(\vartheta )(1+\cos (\vartheta ))\mathrm{d}\vartheta ).$$

(111)

The integrals in equations (110) and (111) are calculated numerically. Vacuum and BT are chosen as inner and outer dielectric, respectively, with the results for variable ${\gamma }^{\infty }$ and ${\sigma }_{22}^{\infty }$ visualized in Figure 13 for an inclined load with $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ . Firstly, in Figure 13(a), the stress ${\sigma }_{22}^{(2)}$ corresponding to the SIF $K_I^{(2)}=\sqrt{\pi a}{\sigma }_{22}^{(2)}$ and representing a constant crack face load with equivalent SIF as ${\mathbf{t}}^{\mathrm{M}}$ is plotted. It is observed that the absolute value of ${\sigma }_{22}^{(2)}$ is maximal with a magnitude up to $30\mathrm{MPa}$ for vertical electric loads in the case of a closed crack, i.e., ${\sigma }_{22}^{\infty }=0$ . Upon opening of the crack, ${\sigma }_{22}^{(2)}$ generally decreases, thus reducing the crack closing effect. It is particularly noteworthy and probably the most essential finding at this point that $K_I^{\mathrm{C}}=-\sqrt{\pi a}{\sigma }_{22}^{\mathrm{C}}$ significantly underestimates the true SIF $K_I^{(2)}$ as depicted in Figure 13(b). While both SIF are nearly identical for a closed crack in BT, see equation (89), their deviation increases virtually independently of the angle ${\gamma }^{\infty }$ and quasi-linearly with the applied load, amounting to more than $60\%$ for ${\sigma }_{22}^1=50\mathrm{MPa}$ . Consequently, the enormous concentration of traction near the crack tip shown in Figure 9(b), although restricted to a tiny portion of the crack faces, reflects in a distinctly larger SIF compared to uniform loading emanating from the capacitor analogy. In contrast to ${\sigma }_{22}^{(2)}$ , Figure 13(c) reveals that ${\sigma }_{12}^{(2)}$ is identically zero for a closed crack and increases with the applied mechanical load. Considering that the absolute value of ${\sigma }_{12}^{(2)}$ in the parameter range of interest, however, remains in the kPa regime, the mode II loading due to ${\mathbf{t}}^{\mathrm{M}}$ may be disregarded. This fact is further underpinned by the mixed mode ratio $K_{\mathrm{II}}^{(2)}/K_I^{(2)}$ illustrated in Figure 13(d), whose sign is sensitive to the angle ${\gamma }^{\infty }$ , yet its magnitude is negligible either way.

Figure 13.

SIF for BT with $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ : (a) and (c) stresses ${\sigma }_{22}^{(2)}$ and ${\sigma }_{12}^{(2)}$ due to Minkowski tractions $t_2^{\mathrm{M}}$ and $t_1^{\mathrm{M}}$ , respectively, as functions of stress load ${\sigma }_{22}^{\infty }$ and angle of the electric load ${\gamma }^{\infty }$ ; (b) and (d) ratios of true and approximated intensity factors $K_I^{(2)}/K_I^C$ as well as mixed mode ratio $K_{\mathrm{II}}^{(2)}/K_I^{(2)}$ versus ${\sigma }_{22}^{\infty }$ for selected angles ${\gamma }^{\infty }$ .

Finally, a total stress ${\sigma }_{22}^{\mathrm{tot}}={\sigma }_{22}^{\infty }+{\sigma }_{22}^{(2)}$ is introduced, providing the total mode I SIF $K_I=\sqrt{\pi a}{\sigma }_{22}^{\mathrm{tot}}$ , and plotted versus the external load ${\sigma }_{22}^{\infty }$ in Figure 14.

Figure 14.

BT loaded electromechanically with ${\sigma }_{22}^{\infty }$ and $|{\mathbf{E}}^{\infty }|=2\mathrm{kV}/\mathrm{mm}$ : total stress ${\sigma }_{22}^{\mathrm{tot}}={\sigma }_{22}^{\infty }+{\sigma }_{22}^{(2)}$ corresponding to the total SIF $K_I=K_I^{(1)}+K_I^{(2)}$ versus mechanical load ${\sigma }_{22}^{\infty }$ for selected angles ${\gamma }^{\infty }$ .

Depending on the angle of the electric load, the crack tip is under compressive load for values of ${\sigma }_{22}^{\infty }$ up to $20\mathrm{MPa}$ due to the large crack closing effect of the Minkowski tractions. As this effect diminishes with increased crack opening, the stress reduction accounts for roughly $20\%$ in the case of large loads of ${\sigma }_{22}^{\infty }=50\mathrm{MPa}$ .

4. Conclusion

The aim of this work was to assess the electric conditions at dielectric crack faces, particularly in light of the commonly applied capacitor analogy representing an approximate model of opened crack faces and providing Neumann boundary conditions in terms of electric surface charges and Coulombic tractions. In the first instance, revisiting the electrostatic solution of an infinite linear isotropic dielectric with elliptic void subjected to remote electric fields, the deficient inverse of a conformal map involved in the transformation from an auxiliary to the physical plane has been amended by carefully examining the square root function on the complex numbers, finally granting access to solution in the entire plane. Based on this closed-form solution, the validity of the capacitor analogy was investigated, unveiling an insignificant deviation of the approximate from the true charges for small crack openings typical in brittle fracture, independent of the remaining problem parameters. As a consequence, both charges and corresponding DIF represent an appropriate substitute for their true counterparts. By contrast, the Coulombic tractions fail to properly embody the Minkowski tractions near the crack tips where the curvature is non-negligible and thus relative differences between both quantities become vast. As a result, the mode I SIF is significantly overestimated by the capacitor analogy, provided that the dielectric permittivity of the plate is much larger than that of the void, which is the case for many dielectric engineering materials. Most strikingly, the latter insight challenges the applicability of small deformation theory in fracture mechanics of dielectrics building on a closed crack in the undeformed configuration, as the electric field distribution turns out particularly sensitive to crack curvature.