Coherent energetic interfaces accounting for in-plane degradation

Abstract

Interfaces can play a dominant role in the overall response of a body. The importance of interfaces is particularly appreciated at small length scales due to large area to volume ratios. From the mechanical point of view, this scale dependent characteristic can be captured by endowing a coherent interface with its own elastic resistance as proposed by the interface elasticity theory. This theory proves to be an extremely powerful tool to explain size effects and to predict the behavior of nano-materials. To date, interface elasticity theory only accounts for the elastic response of coherent interfaces and obviously lacks an explanation for inelastic interface behavior such as damage or plasticity. The objective of this contribution is to extend interface elasticity theory to account for damage of coherent interfaces. To this end, a thermodynamically consistent interface elasticity theory with damage is proposed. A local damage model for the interface is presented and is extended towards a non-local damage model. The non-linear governing equations and the weak forms thereof are derived. The numerical implementation is carried out using the finite element method and consistent tangents are listed. The computational algorithms are given in detail. Finally, a series of numerical examples is studied to provide further insight into the problem and to carefully elucidate key features of the proposed theory.

Appendices

Appendix 1: Some mathematical relations and derivations

In this section we present the derivation of the balance of forces on the interface and the corresponding weak form. Subsequently, the decoupled form of the interface energy, Piola stress and elasticity tensor are provided. Some useful identities and relations used in the derivations are also given without proof.

1.1 Extended divergence theorem

The extended forms of divergence theorem in the material configuration for a bulk tensor field $\{\bullet \}$ and a tensorial quantity on the interface $\left\{ \overline{\bullet }\right\}$ are

$$\begin{aligned}&\int _{{\mathcal {B}}_0} {\mathrm {Div}}\{\bullet \} \mathrm {d}V = \int _{\partial {\mathcal {B}}_0} \left\{ \bullet \right\} \cdot {\varvec{N}} \mathrm {d}A - \int _{ {\mathcal {I}}_0} \llbracket \bullet \rrbracket \cdot \overline{{\varvec{N}}} \mathrm {d}A\;, \end{aligned}$$

(37)

$$\begin{aligned}&\int _{{\mathcal {I}}_0} \overline{{\mathrm {Div}}}\left\{ \overline{\bullet }\right\} \mathrm {d}A = \int _{\partial {\mathcal {I}}_0\setminus \partial {\mathcal {I}}_0^{\mathrm {N}}} \left\{ \overline{\bullet }\right\} \cdot \widetilde{{\varvec{N}}} \mathrm {d}L\nonumber \\&\quad + \int _{\partial {\mathcal {I}}_0^{\mathrm {N}}} \left\{ \overline{\bullet }\right\} \cdot \widetilde{{\varvec{N}}} \mathrm {d}L- \int _{{\mathcal {I}}_0} \overline{C} \left\{ \overline{\bullet }\right\} \cdot \overline{{\varvec{N}}} \mathrm {d}A \;, \end{aligned}$$

(38)

where the curvature of the interface is denoted by $\overline{C}$. Note that $\partial {\mathcal {I}}_0^{\mathrm {N}}$ is the portion of the interface boundary that intersects with the bulk’s boundary, thus $\partial {\mathcal {I}}_0\setminus \partial {\mathcal {I}}_0^{\mathrm {N}} \cap \partial {\mathcal {B}}_0 = \emptyset$.

1.2 Balance of forces on interface

The global form of the balance of forces both in the bulk and on the interface is given as (see, Javili and Steinmann 2010b, for further details concerning thermomechanical solids with surface energy only)

$$\begin{aligned}&\int _{{\mathcal {B}}_0} {\varvec{B}}^{\mathrm {p}} \mathrm {d}V + \int _{{\mathcal {I}}_0} \overline{{\varvec{B}}}{}^{\mathrm {p}} \mathrm {d}A + \int _{\partial {\mathcal {B}}_0^{\mathrm {N}}} \widehat{{\varvec{B}}}{}^{\mathrm {p}}_{\mathrm {N}} \mathrm {d}A + \int _{\partial {\mathcal {I}}_0^{\mathrm {N}}} \widetilde{{\varvec{B}}}{}^{\mathrm {p}}_{\mathrm {N}} \mathrm {d}L\nonumber \\&\quad + \int _{\partial {\mathcal {B}}_0 \setminus \partial {\mathcal {B}}_0^{\mathrm {N}}} {\varvec{P}} \cdot {\varvec{N}} \mathrm {d}A + \int _{\partial {\mathcal {I}}_0 \setminus \partial {\mathcal {I}}_0^{\mathrm {N}}} \overline{{\varvec{P}}} \cdot \widetilde{{\varvec{N}}} \mathrm {d}L= {\varvec{0}}.\nonumber \\ \end{aligned}$$

(39)

Taking the limit ${\mathcal {B}}_0 \rightarrow \emptyset$, and consequently ${\partial {\mathcal {B}}_0} = {{\mathcal {I}}_0}$, with ${\varvec{N}} = \overline{{\varvec{N}}}$ on ${{\mathcal {I}}_0^+}$, ${\varvec{N}} = -\overline{{\varvec{N}}}$ on ${{\mathcal {I}}_0^-}$, ${\partial {\mathcal {I}}_0^{\mathrm {N}}} = \emptyset$, ${\partial {\mathcal {B}}_0^{\mathrm {N}}} = \emptyset$, and taking into account the extended forms of the divergence theorem (37) and (38), one obtains the local balance of forces on the interface as

$$\begin{aligned}&\int _{\partial {\mathcal {I}}_0} \overline{{\varvec{P}}} \cdot \widetilde{{\varvec{N}}} \mathrm {d}L + \int _{{\mathcal {I}}_0 } {\varvec{P}} \cdot {\varvec{N}} \mathrm {d}A + \int _{{\mathcal {I}}_0} \overline{{\varvec{B}}}{}^{\mathrm {p}} \mathrm {d}A = {\varvec{0}} \nonumber \\&\quad \Longrightarrow \int _{{\mathcal {I}}_0} \overline{{\mathrm {Div}}}~\overline{\varvec{P}} + \llbracket \varvec{P}\rrbracket \cdot \overline{{\varvec{N}}} + \overline{{\varvec{B}}}{}^{\mathrm {p}} \mathrm {d}A = {\varvec{0}}\;. \end{aligned}$$

(40)

From arbitrariness of ${\mathcal {B}}_0$ and thus ${\mathcal {I}}_0$, the balance of force on the interface listed in Table 2 then follows. In the case that the interface is not energetic i.e. $\overline{\varvec{P}} = {\varvec{0}}$, and in the absence of interface body force ($\overline{{\varvec{B}}}{}^{\mathrm {p}} = {\varvec{0}}$), the classical traction continuity condition is recovered.

1.3 Weak form of the balance of forces

The localized balance equations in the bulk and on the interface, given in Table 2 are tested from the left with vector valued functions $\delta {\varvec{\varphi }}$ and $\delta \overline{{\varvec{\varphi }}}$, respectively as follows

$$\begin{aligned}&\int _{{\mathcal {B}}_0} \delta {\varvec{\varphi }} \cdot [~ {\mathrm {Div}}{\varvec{P}} + {\varvec{B}}^{\mathrm {p}} ~] \mathrm {d}V \nonumber \\&\quad + \int _{{\mathcal {I}}_0} \delta \overline{{\varvec{\varphi }}} \cdot [~ \overline{{\mathrm {Div}}}\, \overline{{\varvec{P}}} + \overline{{\varvec{B}}}{}^{\mathrm {p}} + \llbracket {\varvec{P}}\rrbracket \cdot \overline{{\varvec{N}}} ~] \mathrm {d}A = 0,\nonumber \\ \end{aligned}$$

(41)

which can be alternatively written as

$$\begin{aligned}&\int _{{\mathcal {B}}_0} -{\varvec{P}} : {\mathrm {Grad}}\delta {\varvec{\varphi }} + {\mathrm {Div}}( \delta {\varvec{\varphi }} \cdot {\varvec{P}}) + \delta {\varvec{\varphi }} \cdot {\varvec{B}}^{\mathrm {p}} \mathrm {d}V \nonumber \\&\quad + \int _{{\mathcal {I}}_0} -\overline{{\varvec{P}}} : \overline{{\mathrm {Grad}}}\delta \overline{{\varvec{\varphi }}} + \overline{{\mathrm {Div}}}( \delta \overline{{\varvec{\varphi }}} \cdot \overline{{\varvec{P}}}) \mathrm {d}A\nonumber \\&\quad + \int _{{\mathcal {I}}_0} \delta \overline{{\varvec{\varphi }}} \cdot \overline{{\varvec{B}}}{}^{\mathrm {p}} + \delta \overline{{\varvec{\varphi }}} \cdot [~\llbracket {\varvec{P}}\rrbracket \cdot \overline{{\varvec{N}}} ~] \mathrm {d}A = 0\;, \end{aligned}$$

(42)

and using the extended forms of divergence theorem (37) and (38), for various parts of the body results in

$$\begin{aligned}&\int _{{\mathcal {B}}_0} {\varvec{P}} : {\mathrm {Grad}}\delta {\varvec{\varphi }} \mathrm {d}V - \int _{\partial {\mathcal {B}}_0^{\mathrm {N}}} \delta {\varvec{\varphi }} \cdot [~ {\varvec{P}} \cdot {\varvec{N}} ~] \mathrm {d}A \nonumber \\&\quad + \int _{{\mathcal {I}}_0} \llbracket \delta {{\varvec{\varphi }}} \cdot {\varvec{P}}\rrbracket \cdot \overline{{\varvec{N}}}\mathrm {d}A - \int _{{\mathcal {B}}_0} \delta {\varvec{\varphi }} \cdot {\varvec{B}}^{\mathrm {p}} \mathrm {d}V \nonumber \\&\quad + \int _{{\mathcal {I}}_0} \overline{{\varvec{P}}} : \overline{{\mathrm {Grad}}}\delta \overline{{\varvec{\varphi }}}\mathrm {d}A - \int _{\partial {\mathcal {I}}_0^{{\mathrm {N}}}} \delta \overline{{\varvec{\varphi }}} \cdot [~\overline{{\varvec{P}}} \cdot \widetilde{{\varvec{N}}}~]\mathrm {d}L\nonumber \\&\quad - \int _{{\mathcal {I}}_0} \delta \overline{{\varvec{\varphi }}} \cdot \overline{{\varvec{B}}}{}^{\mathrm {p}} + \delta \overline{{\varvec{\varphi }}} \cdot [~\llbracket {\varvec{P}}\rrbracket \cdot \overline{{\varvec{N}}} ~] \mathrm {d}A = 0 . \end{aligned}$$

(43)

On the Neumann boundaries of the bulk and interface, ${\varvec{P}} \cdot {\varvec{N}} = \widehat{{\varvec{B}}}_{\mathrm {N}}^{\mathrm {p}}$ and $\overline{{\varvec{P}}} \cdot \widetilde{{\varvec{N}}} = \widetilde{{\varvec{B}}}{}^{\mathrm {p}} _{\mathrm {N}}$, respectively. Noting , $\llbracket \delta {{\varvec{\varphi }}} \rrbracket = {\varvec{0}}$, for coherent interfaces, and , Eq. (43) simplifies to the weak form Eq. (18).

1.4 Decoupled form of stress and elasticity tensor

It is sometimes useful to decouple the bulk deformation into the volumetric and isochoric part. In analogy, the volumetric⁶ and isochoric part of the interface deformation read

$$\begin{aligned}&\overline{{\varvec{F}}} = \overline{{\varvec{F}}}{}^\text {vol}\overline{{\varvec{F}}}{}^\text {iso}\quad \text {with}\quad \overline{{\varvec{F}}}{}^\text {vol}= \overline{J}{}^{1/2}\overline{{\varvec{I}}}\nonumber \\&\quad \text {and}\quad \overline{{\varvec{F}}}{}^\text {iso}= \overline{J}{}^{-1/2} \overline{{\varvec{F}}} \;. \end{aligned}$$

(44)

Furthermore, the Helmholtz energy can be written as

$$\begin{aligned} \overline{\varPsi }\left( \overline{{\varvec{F}}}\right) = \overline{\varPsi }\left( \overline{{\varvec{F}}}{}^\text {iso},\,\overline{{\varvec{F}}}{}^\text {vol}\right) \;. \end{aligned}$$

(45)

The Piola stress reads

$$\begin{aligned} \begin{aligned}&\overline{{\varvec{P}}} {:}{=} \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}} = \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}}:\frac{\partial \overline{{\varvec{F}}}{}^\text {iso}}{\partial \overline{{\varvec{F}}}} + \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {vol}}:\frac{\partial \overline{{\varvec{F}}}{}^\text {vol}}{\partial \overline{{\varvec{F}}}}&\text {or}&\\&\overline{{\varvec{P}}} = \overline{{\varvec{P}}}{}^\text {iso}+\overline{{\varvec{P}}}{}^\text {vol}\;, \end{aligned} \end{aligned}$$

(46)

with

$$\begin{aligned} \overline{{\varvec{P}}}{}^\text {vol}= & {} \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {vol}} : \frac{\partial \overline{{\varvec{F}}}{}^\text {vol}}{\partial \overline{{\varvec{F}}}} = \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {vol}} : \left[ \frac{1}{2}\overline{J}{}^{1/2} \overline{{\varvec{I}}} \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] \;, \end{aligned}$$

(47)

$$\begin{aligned} {\overline{{\varvec{P}}}}{}^\text {iso}= & {} \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}} : \frac{\partial \overline{{\varvec{F}}}{}^\text {iso}}{\partial \overline{{\varvec{F}}}} = \overline{J}{}^{-1/2} \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}} :\left[ \overline{{\mathbb {I}}}-\displaystyle \frac{1}{2} \overline{{\varvec{F}}} \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] .\nonumber \\ \end{aligned}$$

(48)

The Piola stress tangent follows

$$\begin{aligned} \overline{{\mathbb {A}}} {:=} \frac{\partial \overline{{\varvec{P}}}}{\partial \overline{{\varvec{F}}}} = \frac{\partial \overline{{\varvec{P}}}{}^\text {iso}}{\partial \overline{{\varvec{F}}}} + \frac{\partial \overline{{\varvec{P}}}{}^\text {vol}}{\partial \overline{{\varvec{F}}}}&\text {or}&\overline{{\mathbb {A}}} = \overline{{\mathbb {A}}}{}^\text {iso}+ \overline{{\mathbb {A}}}{}^\text {vol}\;, \end{aligned}$$

(49)

with

$$\begin{aligned} \overline{{\mathbb {A}}}{}^\text {vol}= & {} \frac{\partial \overline{{\varvec{P}}}{}^\text {vol}}{\partial \overline{{\varvec{F}}}} = \frac{1}{4}\overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\otimes \left[ \left[ \frac{\partial ^2\overline{\varPsi }}{\partial \overline{{\varvec{F}}}_\text {vol}\partial \overline{{\varvec{F}}}_\text {vol}} :\left[ \overline{{\varvec{I}}}\otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}} \right] \right] :\overline{{\varvec{I}}} \right] \nonumber \\&- \frac{1}{4}\overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\otimes \left[ \overline{J}{}^{-1/2}\frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}_\text {vol}}:\left[ \overline{{\varvec{I}}}\otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] \right] \nonumber \\&+\frac{1}{2}\overline{J}{}^{-1/2}\left[ \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}_\text {vol}} :\overline{{\varvec{I}}}\right] \overline{{\mathbb {D}}}\;, \end{aligned}$$

(50)

$$\begin{aligned} \overline{{\mathbb {A}}}{}^\text {iso}= & {} \frac{\partial \overline{{\varvec{P}}}{}^\text {iso}}{\partial \overline{{\varvec{F}}}}= -\frac{1}{2}\overline{J}{}^{-1/2}\left[ \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}} : \left[ \overline{{\mathbb {I}}}-\displaystyle \frac{1}{2} \overline{{\varvec{F}}} \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] \right] \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}} \nonumber \\&+ \overline{J}{}^{-1/2} \left[ \overline{{\mathbb {A}}}_1 - \frac{1}{2}\left[ \overline{{\mathbb {A}}}_2 + \overline{{\mathbb {A}}}_3 \right] \right] \;, \end{aligned}$$

(51)

where

$$\begin{aligned}&\overline{{\mathbb {A}}}_1 = \left[ \overline{{\varvec{I}}} ~ \overline{\otimes }~ {\varvec{i}}\right] :\frac{\partial ^2 \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}\partial \overline{{\varvec{F}}}{}^\text {iso}}:\left[ \overline{{\mathbb {I}}}-\displaystyle \frac{1}{2} \overline{{\varvec{F}}} \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] \;,\nonumber \\&\overline{{\mathbb {A}}}_2 = \left[ \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}}: \overline{{\varvec{F}}}\right] \overline{{\mathbb {D}}}\;,\nonumber \\&\overline{{\mathbb {A}}}_3 = \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\otimes \left[ \frac{\partial \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}}:\overline{{\mathbb {I}}} + \frac{\partial ^2 \overline{\varPsi }}{\partial \overline{{\varvec{F}}}{}^\text {iso}\partial \overline{{\varvec{F}}}{}^\text {iso}} :\left[ \overline{{\mathbb {I}}}-\displaystyle \frac{1}{2} \overline{{\varvec{F}}} \otimes \overline{{\varvec{F}}}{}^{-{\mathrm {t}}}\right] : \overline{{\varvec{F}}}\right] \;. \end{aligned}$$

(52)

Appendix 2: Differential geometry of two-dimensional manifolds embedded in three-dimensional space

In this section we briefly review some common terminologies in differential geometry of two-dimensional manifolds frequently used in this work to represent the interface elasticity theory. Finding the shortest distance (minimal geodesic) on a curved two-dimensional manifold (interface) embedded in a three-dimensional Euclidean space is presented subsequently, which is employed in the calculation of the non-local coefficients on the interface.

Fig. 16

Illustration of the Cartesian basis ${\mathcal {E}}$ and an arbitrary basis ${\mathcal {A}}$ (a), and a cylindrical interface with its minimal geodesic (helix) along with the polar coordinate system (b). The red portion of the helix on the cylinder is the shortest arc-length of all the curves connecting the point $\overline{\varvec{x}}_\text {r}$ and $\overline{\varvec{x}}_\text {s}$. In (a), the ${\varvec{a}}_1$ and ${\varvec{a}}_2$ in ${\mathcal {A}}$ are not necessarily orthogonal to each other, whereas the Cartesian coordinate system ${\mathcal {E}}$ is composed of three orthogonal axes. In (b), every point on the surface of the cylinder is characterized by the coordinates $\rho$ the radius, $\theta$ the sweeping angle and z the height

A parametric interface ${\mathscr {I}}$ in ${\mathbb {E}}^3$ (three-dimensional embedding Euclidean space) is a map ${\mathscr {I}} : {\mathcal {I}} \rightarrow {\mathbb {E}}^3$ (with ${\mathcal {I}} \in {\mathbb {E}}^2$) such that its differential has rank 2 at all points $\eta ^\alpha \in {\mathcal {I}}$ with $\alpha = 1 \,,\, 2$. The interface can be defined by a parametric equation $\overline{{\varvec{x}}} : {\mathcal {I}} \rightarrow {\mathbb {E}}^3$ as $\overline{{\varvec{x}}} = \overline{{\varvec{x}}}(\eta ^\alpha )$, where $\overline{{\varvec{x}}}$ is a vector-valued function of the scalar-valued parameters $\eta ^\alpha$. The tangent space to $\overline{{\varvec{x}}}$ at $\eta ^\alpha$ is the linear map $\mathrm {D}\overline{{\varvec{x}}}(\eta ^\alpha ) : {\mathbb {E}}^2 \rightarrow {\mathbb {E}}^3$ denoted by $T {\mathscr {I}}$ where $\mathrm {D}\overline{{\varvec{x}}}(\eta ^\alpha )$ is the differential of the interface. The tangent vectors ${\varvec{a}}_\alpha \in T {\mathscr {I}}$, i.e. the covariant interface basis vectors (see Fig. 16a) are then given by ${\varvec{a}}_\alpha = \partial _{\eta ^\alpha } \overline{{\varvec{x}}}(\eta ^\alpha )$.

The corresponding contravariant (dual) interface basis vectors ${\varvec{a}}^\alpha$ are related to the covariant interface basis vectors by means of the co- and contravariant interface metric coefficients $a_{\alpha \beta }$ (first fundamental form for the interface) and $a^{\alpha \beta }$, respectively, as

$$\begin{aligned}&{\varvec{a}}^\alpha = a^{\alpha \beta }{\varvec{a}}_\beta \qquad \text {and}\qquad {\varvec{a}}_\alpha = a_{\alpha \beta }{\varvec{a}}^\beta \qquad \text {where}\nonumber \\&a_{\alpha \beta } = {\varvec{a}}_{\alpha }\cdot {\varvec{a}}_{\beta } \qquad \text {and} \qquad a^{\alpha \beta } = {\varvec{a}}^{\alpha } \cdot {\varvec{a}}^{\beta } \;. \end{aligned}$$

(53)

Note that the two metrics are inverse to each other, i.e. $[a^{\alpha \beta }] = [a_{\alpha \beta }]^{-1}$. The contra- and covariant base vectors ${\varvec{a}}_3$ and ${\varvec{a}}^3$, normal to $T{\mathscr {I}}$, are defined by

$$\begin{aligned} {\varvec{a}}^3 {:=} {\varvec{a}}_1 \times {\varvec{a}}_2&\text {and}&{\varvec{a}}_3 {:=} [a^{33}]^{-1} {\varvec{a}}^3\;, \end{aligned}$$

(54)

such that ${\varvec{a}}^3 \cdot {\varvec{a}}_3 = 1$. Correspondingly the unit normal to the interface, parallel to ${\varvec{a}}^3$ and ${\varvec{a}}_3$, can be calculated as $\overline{{\varvec{n}}} = {\varvec{a}}_3/|{\varvec{a}}_3| = {\varvec{a}}^3/|{\varvec{a}}^3|$. The interface identity tensor is defined as $\overline{{\varvec{i}}} {:=} {\varvec{i}} - {\varvec{a}}_3 \otimes {\varvec{a}}^3 = {\varvec{i}} - \overline{{\varvec{n}}} \otimes \overline{{\varvec{n}}}$, where ${\varvec{i}}$ denotes the ordinary mixed-variant unit tensor of the embedding Euclidean space. The interface gradient, divergence and determinant operators in a general curvilinear coordinate are defined as

$$\begin{aligned}&\overline{{\mathrm {grad}}}\left\{ \overline{\bullet }\right\} {:=}\frac{\partial \left\{ \overline{\bullet }\right\} }{\partial \eta ^\alpha }\otimes {\varvec{a}}_\alpha , \quad \overline{{\mathrm {div}}}\left\{ \overline{\bullet }\right\} {:=}\frac{\partial \left\{ \overline{\bullet }\right\} }{\partial \eta ^\alpha }\cdot {\varvec{a}}_\alpha&\text {and}&\nonumber \\&\overline{{\mathrm {det}}}\left\{ \overline{\bullet }\right\} {:=}\frac{\left[ \left\{ \overline{\bullet }\right\} \cdot {\varvec{a}}_1\right] \times \left[ \left\{ \overline{\bullet }\right\} \cdot {\varvec{a}}_2\right] }{|{\varvec{a}}_1\times {\varvec{a}}_2|}\;. \end{aligned}$$

(55)

Having obtained the normal to the interface, co- and contravariant basis vectors, the interface curvature tensor $\overline{{\varvec{k}}}$, second-order superficial deformation gradient tensor $\overline{{\varvec{F}}}$ and its inverse $\overline{{\varvec{f}}}$ are defined, respectively, as

$$\begin{aligned}&\overline{{\varvec{k}}} {:=} -\overline{{\mathrm {grad}}}\overline{{\varvec{n}}},\qquad \overline{{\varvec{F}}} {:=} \overline{{\mathrm {Grad}}}\overline{{\varvec{x}}} = {\varvec{a}}_\alpha \otimes {\varvec{A}}^\alpha&\text {and}&\nonumber \\&\overline{{\varvec{f}}} {:=} \overline{{\mathrm {grad}}}\overline{{\varvec{X}}} = {\varvec{A}}_\alpha \otimes {\varvec{a}}^\alpha \;. \end{aligned}$$

(56)

Note that due to the superficiality (rank deficiency) of the interface deformation gradient $\overline{{\varvec{F}}}$, its inverse $\overline{{\varvec{f}}}$ must be computed using Eq. (56)$_3$.

Table 6 $L_2$ norm of the residual for three scenarios: bulk and interface damaged, only bulk damaged and only interface damaged

The geodesics are the general form of straight lines when applied to curved, three-dimensional interfaces. The minimal geodesics in differential geometry are the shortest distance paths between two points on an interface. Clearly, minimal geodesics on the interfaces are curves of minimum arc-lengths. To find the minimal geodesics, first we introduce the parameter t on which the interface parameters $\eta ^\alpha$ are dependent. The arc-length of the curve connecting any two points $\overline{{\varvec{x}}}_\text {r}(t_1)$ and $\overline{{\varvec{x}}}_\text {s}(t_2)$ on the curved interface can be written as

$$\begin{aligned} I = \int _{t_1}^{t_2} \sqrt{a_{\alpha \beta }\,\frac{\mathrm {d}\eta ^\alpha (t)}{\mathrm {d}t}\,\frac{\mathrm {d}\eta ^\beta (t)}{\mathrm {d}t}} ~\mathrm {d}t \;, \end{aligned}$$

(57)

where the integrand is the infinitesimal line element. Next, to find the shortest arc-length of the curves connecting the two points (minimal geodesic), the functional I is minimized. In doing so, the integrand in Eq. (57) (the Lagrangian), denoted by $L(\eta ^\alpha (t), \dot{\eta }^\alpha (t))$ must satisfy the Euler-Lagrange equations of the form

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\frac{\partial L}{\partial \dot{\eta }^\alpha } - \frac{\partial L}{\partial \eta ^\alpha }=0 \;, \end{aligned}$$

(58)

where $\dot{\{\bullet \}}$ signifies a differentiation with respect to the parameter t.

Convergence behavior and run-time analysis

In this section we present some data on the convergence behavior and run-time of the computational problem at hand. Firstly, the $L_2$ norms of the residual of few increments for all the three examples discussed in Sect. 6 are given in Table 6. As mentioned before, due to the consistent linearization, the asymptotic quadratic rate of convergence associated with the Newton–Raphson scheme is achieved. Secondly, to study how the interactive radius would change the memory consumption and run-time, we only allow the damage initiation in the bulk, increase both number of elements⁷ and interactive radius R and measure the run-time for one iteration per increment. These measurements are carried out on a machine with the following specifications:

• processors: Intel Core i7-4770 CPU 3.40GHz $\times$ 8,

• memory: 15.6GB,

• OS type: 64-bit.

The run-time measurements per iteration together with the memory usage for a serial and parallel code are given in Table 7. The parallel implementation is carried out using the MPI library. Note that the total run-time for a mesh size of $100\times 100\times 1$ with $R=0.01$ mm is approximately 39.3 h. A parallel implementation, using 4 processors, results in a speed-up of 3.5 and consequently a run-time of 7.75 h. As indicated by the presented data in Table 7, increasing the interactive radius R causes a substantial increase in the run-time. The main source of the time consumption, as expected, is in the stiffness assembly since firstly the non-localization is integral-type and secondly the linearization is consistent (introducing the double integrals in the stiffness formulation). However, as mentioned in Jirásek and Patzák (2002), if this type of non-localization is chosen, the extra time spent on the stiffness assembly due to consistent linearization in every iteration might be compensated by fewer iterations required to meet the convergence criterion. This matter becomes critical if a more stringent convergence criterion is necessary. For further details on the implementation issues of the consistent linearization of non-local damage problems of integral-type we refer to Jirásek and Patzák (2002).

Table 7 Serial- and parallel-code run-time measurements per iteration together with the memory consumption for various mesh sizes and values of interactive radius