# American Institute of Mathematical Sciences

December  2017, 10(6): 1539-1561. doi: 10.3934/dcdss.2017079

## An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law

 1 Technical University of Košice, Civil Engineering Faculty, Vysokoškolská 9,042 00 Košice, Slovakia 2 University of Seville, School of Engineering, Camino de los Descubrimientos s/n, 41092 Seville, Spain

Dedicated to Tomáš Roubíček on the occasion of his 60th birthday

Received  October 2016 Revised  February 2017 Published  June 2017

A new quasi-static and energy based formulation of an interface damage model which provides interface traction-relative displacement laws like in traditional trilinear (with bilinear softening) or generally multilinear cohesive zone models frequently used by engineers is presented. This cohesive type response of the interface may represent the behaviour of a thin adhesive layer. The level of interface adhesion or damage is defined by several scalar variables suitably defined to obtain the required traction-relative displacement laws. The weak solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive double minimization in displacements and damage variables separately. The symmetric Galerkin boundary-element method is applied for the spatial discretization. Sequential quadratic programming is implemented to resolve each partial minimization in the recursive scheme applied to compute the time-space discretized solutions. Sample 2D numerical examples demonstrate applicability of the proposed model.

Citation: Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079
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##### References:
The used notation for two bonded domains.
Stress-displacement curves for (a) the bilinear and (b) multilinear CZMs with $m_{\rm d}{=}3$.
Stress-displacement relations for multilinear CZMs with $m_{\rm d}{=}3$: (a) Mode 'subsequent', (b) Mode 'at once'.
An example of stress-relative displacement relation for a multilinear CZM with $m_{\rm d}{=}3$, $\sigma_{\max\,{\rm n}}{=}1.25\sigma_{\max\,{\rm t}}$, $G_{\tiny\rm IIc}{=}2G_{\tiny\rm Ic}$.
Simple tension in the two-square example, $a_1{=}200\,$mm: (a) the problem layout, (b) the traction-relative displacement law in the cohesive zone, (c) the loading function $g$ from (24).
The stress-displacement relation at the point $x_2{=}50$mm (the quarter of the interface) and the evolution of the damage parameters at the same point: (a) Mode 'subsequent', (b) Mode 'at once'.
Double cantilever beam: (a) the problem layout: $\ell{=}190\,$mm, $\ell_{\text{ini}}{=}55\,$mm, $w{=}20\,$mm, $h{=}5\,$mm, (b) the traction-relative displacement law in the cohesive zone: $u_0{=}0.014\,$mm, $u_1{=}0.25\,$mm, $u_{\text c}{=}4\,$mm, $\sigma_0{=}62\,$MPa, $\sigma_1{=}0.67\,$MPa.
Deformations of DCB, the damage evolution and normal stress distribution in the partially cracked interface at selected time instants corresponding to prescribed displacement $g$, $\ell_{\rm ini}$ is the initial crack length, cf. Figure 7.
Normal stress-relative displacement graphs at the interface point $x_1{=}\ell_{\text{ini}}{+}4$mm. The damage range is kept the same in the right and left part, only the range for the normal stress $\sigma_{\rm n}$ is changed in the right picture.
The applied forces for DCB calculated at the place where the vertical displacement is imposed.
The mixed mode beam, cf. [40]; (a) the problem layout: $\ell{=}120$mm, $\ell_{\text C}{=}92$mm, $\ell_{\text{ini}}{=}8$mm, $h{=}20$mm, $w{=}2$mm, $s{=}2$mm, (b) used stress-displacement law in the cohesive zone: $u_1{=}2u_0$, $u_2{=}3u_0$, $u_{\text c}{=}4u_0$, $\sigma_2{=}\frac18\sigma_0$ and $\sigma_0{=}7.5\,$MPa, where $u_0{=}0.01\,$mm in normal component; in the tangential component either the same value (no dependence on mode-mixity) or $u_0{=}0.04\,$mm (mixed-mode dependent: $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$).
The total reaction forces for the mixed-mode beam calculated at the simple support constraint: (a) observing the influences of the fracture mode mixity ($G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$ or $G_{\tiny\rm IIc}{=}G_{\tiny\rm Ic}$) and viscosity (solid lines for no viscosity ${\tau_{\rm r}}{=}0$, dashed lines for ${\tau_{\rm r}}{=}10$ms), (b) changes of the solution for various discretizations.
The energy evolution and fulfillment of the energy balance (7) for various discretizations calculated for the mixed-mode beam, $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$: (a) no viscosity ${\tau_{\rm r}}{=}0$, (b) viscosity with ${\tau_{\rm r}}{=}10$ms.
Deformations of the beam, the damage evolution and stress distribution in the cracked interface at selected time instants corresponding to the prescribed displacement $g$: comparison of the cases $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$ or $G_{\tiny\rm IIc}{=}G_{\tiny\rm Ic}$ (referenced respectively by indices 4 and 1) with no viscosity.
Deformations of the beam, the damage evolution and stress distribution in the cracked interface at selected time instants corresponding to the prescribed displacement $g$: comparison of the cases ${\tau_{\rm r}}{=}10$ms and ${\tau_{\rm r}}{=}0$ (referenced respectively by indices 10 and 0) with $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$.
Stress-relative displacement graphs at selected points of the interface: ${\tau_{\rm r}}{=}0$, $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$.
Stress-relative displacement graphs at selected points of the interface: ${\tau_{\rm r}}{=}0$, $G_{\tiny\rm IIc}{=}G_{\tiny\rm Ic}$.
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