Non-coherent energetic interfaces accounting for degradation

Abstract

Within the continuum mechanics framework, there are two main approaches to model interfaces: classical cohesive zone modeling (CZM) and interface elasticity theory. The classical CZM deals with geometrically non-coherent interfaces for which the constitutive relation is expressed in terms of traction–separation laws. However, CZM lacks any response related to the stretch of the mid-plane of the interface. This issue becomes problematic particularly at small scales with increasing interface area to bulk volume ratios, where interface elasticity is no longer negligible. The interface elasticity theory, in contrast to CZM, deals with coherent interfaces that are endowed with their own energetic structures, and thus is capable of capturing elastic resistance to tangential stretch. Nonetheless, the interface elasticity theory suffers from the lack of inelastic material response, regardless of the strain level. The objective of this contribution therefore is to introduce a generalized mechanical interface model that couples both the elastic response along the interface and the cohesive response across the interface whereby interface degradation is taken into account. The material degradation of the interface mid-plane is captured by a non-local damage model of integral-type. The out-of-plane decohesion is described by a classical cohesive zone model. These models are then coupled through their corresponding damage variables. 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 derived. 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.

Appendix: Some mathematical relations and derivations

In this section we derive the weak form of the balance of forces. Some useful identities and relations used in the derivations are also given without proof.

1.1 Extended divergence theorem

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

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

(42)

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

(43)

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 and moments on the interface

The global form of the balance of forces both in the bulk and on the interface is given as [see [45], for further details concerning thermomechanical solids with surface energy only]

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

(44)

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 (42) and (43), one obtains the local balance of forces on the interface as

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

(45)

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.

The global form of balance of moment in the absence of body forces reads

$$\begin{aligned} \int _{\partial \mathcal {B}_0} {\varvec{r}} \times [{\varvec{P}} \cdot {\varvec{N}}] \mathrm {d}A + \int _{\partial \mathcal {I}_0} \overline{{\varvec{r}}} \times [\overline{{\varvec{P}}}\cdot \widetilde{{\varvec{N}}}]\mathrm {d}L = \varvec{0}, \end{aligned}$$

(46)

where ${\varvec{r}}$ and $\overline{{\varvec{r}}}$ are the bulk and interface position vectors. Now by using the extended divergence theorem and the relations

$$\begin{aligned}&\int _{\partial \mathcal {B}_0} {\varvec{r}}\times {\varvec{P}} \cdot {\varvec{N}} \,\mathrm {d}A = \int _{\mathcal {B}_0} {\varvec{r}} \times \text {Div} {\varvec{P}} + {\varvec{\varepsilon }} :{\varvec{F}} \cdot {\varvec{P}}^\mathrm {t} \,\mathrm {d}V \quad \text {and} \nonumber \\&\quad \int _{\partial \mathcal {I}_0} \overline{{\varvec{r}}}\times \overline{{\varvec{P}}} \cdot \overline{{\varvec{N}}} \,\mathrm {d}L = \int _{\mathcal {I}_0} \overline{{\varvec{r}}} \times \overline{\text {Div}} \overline{{\varvec{P}}} + {\varvec{\varepsilon }} : \overline{{\varvec{F}}} \cdot \overline{{\varvec{P}}}{}^\mathrm {t} \,\mathrm {d}A,\nonumber \\ \end{aligned}$$

(47)

one obtains

$$\begin{aligned}&\int _{\mathcal {B}_0} {\varvec{r}} \times \text {Div} {\varvec{P}} + {\varvec{\varepsilon }} :{\varvec{F}}^\mathrm {t} \cdot {\varvec{P}} \,\mathrm {d}V + \int _{\mathcal {I}_0} {{\varvec{r}}} \times \llbracket {\varvec{P}}\rrbracket \cdot \overline{{\varvec{N}}} \,\mathrm {d}A \nonumber \\&\quad + \int _{\mathcal {I}_0} \overline{{\varvec{r}}} \times \overline{\text {Div}} \overline{{\varvec{P}}} + {\varvec{\varepsilon }} : \overline{{\varvec{F}}} \cdot \overline{{\varvec{P}}}{}^\mathrm {t} \,\mathrm {d}A = \varvec{0}. \end{aligned}$$

(48)

By taking the limit $\mathcal {B}_0 \rightarrow \emptyset$, noting that , and using the balance of forces on the interface, listed in Table 2, we find

(49)

which simplifies to

(50)

Now by noting we have

(51)

Note that since in this work the cohesive traction ${\varvec{T}}$ is co-linear with the displacement jump vector $\llbracket {\varvec{r}} \rrbracket$, the balance of moments on the interface becomes $\varvec{\varepsilon }: \overline{{\varvec{F}}} \cdot \overline{{\varvec{P}}}{}^\mathrm {t} = \varvec{0}$, thus $\overline{{\varvec{P}}} \cdot \overline{{\varvec{F}}}{}^\mathrm {t} = \overline{{\varvec{F}}} \cdot \overline{\varvec{P}}{}^\mathrm {t}$.

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

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

(52)

which can be alternatively written as

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

(53)

and using the extended forms of divergence theorem (42) and (43), for various parts of the body results in

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

(54)

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 , Eq. (54) takes the form

(55)

Since $\llbracket \delta {{\varvec{\varphi }}} \rrbracket \ne \varvec{0}$, for non-coherent interfaces, and , Eq. (55) simplifies to the weak form Eq. (32).