# 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 and Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079
##### References:

show all references

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

##### 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: ${\tau_{\rm r}}{=}0$, $G_{\tiny\rm IIc}{=}G_{\tiny\rm Ic}$.
 [1] Marita Thomas, Chiara Zanini. Cohesive zone-type delamination in visco-elasticity. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1487-1517. doi: 10.3934/dcdss.2017077 [2] G. Leugering, Marina Prechtel, Paul Steinmann, Michael Stingl. A cohesive crack propagation model: Mathematical theory and numerical solution. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1705-1729. doi: 10.3934/cpaa.2013.12.1705 [3] Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065 [4] Tomáš Roubíček, V. Mantič, C. G. Panagiotopoulos. A quasistatic mixed-mode delamination model. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 591-610. doi: 10.3934/dcdss.2013.6.591 [5] Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012 [6] Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075 [7] Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1657-1686. doi: 10.3934/cpaa.2013.12.1657 [8] Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027 [9] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [10] Tien-Tsan Shieh. From gradient theory of phase transition to a generalized minimal interface problem with a contact energy. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729 [11] Christopher J. Larsen. Local minimality and crack prediction in quasi-static Griffith fracture evolution. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 121-129. doi: 10.3934/dcdss.2013.6.121 [12] Michael Stiassnie, Raphael Stuhlmeier. Progressive waves on a blunt interface. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3171-3182. doi: 10.3934/dcds.2014.34.3171 [13] Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807 [14] Qing Liu, Bingo Wing-Kuen Ling, Qingyun Dai, Qing Miao, Caixia Liu. Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1993-2011. doi: 10.3934/jimo.2020055 [15] Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1185-1203. doi: 10.3934/dcdsb.2012.17.1185 [16] Frédéric Lebon, Raffaella Rizzoni. Modeling a hard, thin curvilinear interface. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1569-1586. doi: 10.3934/dcdss.2013.6.1569 [17] Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196 [18] Leszek Gasiński, Piotr Kalita. On dynamic contact problem with generalized Coulomb friction, normal compliance and damage. Evolution Equations and Control Theory, 2020, 9 (4) : 1009-1026. doi: 10.3934/eect.2020049 [19] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 [20] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulations of a frictional contact problem with damage and memory. Mathematical Control and Related Fields, 2022, 12 (3) : 621-639. doi: 10.3934/mcrf.2021037

2021 Impact Factor: 1.865