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
References:
[1]

L. Banks-Sills and D. Askenazi, A note on fracture criteria for interface fracture, Int. J. Fracture, 103 (2000), 177-188. Google Scholar

[2]

O. Barani, M. Mosallanejad and S. Sadrnejad, Fracture analysis of cohesive soils using bilinear and trilinear cohesive laws, Int. J. Geomech., 04015088,2015.Google Scholar

[3]

Z. P. Bažant and J. Planas, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, 1998.Google Scholar

[4]

J. Besson, G. Cailletaud, J. L. Chaboche and S. Forest, Non-Linear Mechanics of Materials, Springer, Dordrecht, 2010.Google Scholar

[5]

W. Brocks, A. Cornec and I. Scheider, Computational aspects of Nonlinear Fracture Mechanics (Chapter 3. 03), In: Numerical and Computational Methods (Vol. 3), R. de Borst, H. A. Mang (Vol. Eds. ), Comprehensive Structural: Fracture of Materials from Nano to Macro, I. Milne, R. O. Ritchie, B. Karihaloo (Eds. ), pp. 127{209, Elsevier, 2003.Google Scholar

[6]

A. Carpinteri, Post-peak and post-bifurcation analysis of catastrophic softening behaviour (snap-back instability), Engineering Fracture Mechanics, 32 (1989), 265-278. Google Scholar

[7]

C. G. DávilaC. A. Rose and P. P. Camanho, A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture, International Journal of Fracture, 158 (2009), 211-223. Google Scholar

[8]

C. G. Dávila, C. A. Rose and E. V. Iarve, Modeling fracture and complex crack networks in laminated composites, in Mathematical Methods and Models in Composites, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 297-347. doi: 10.1142/9781848167858_0008. Google Scholar

[9]

G. Del Piero and M. Raous, A unified model for adhesive interfaces with damage, viscosity, and friction, Europ. J. of Mechanics A/Solids, 29 (2010), 496-507. doi: 10.1016/j.euromechsol.2010.02.004. Google Scholar

[10]

Z. Dostál, Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications, Springer, Berlin, 2009. Google Scholar

[11]

M. Frémond, Dissipation dans l'adherence des solides, C.R. Acad. Sci., Paris, Sér.Ⅱ, 300 (1985), 709-714. Google Scholar

[12]

G. V. GuineaJ. Planas and M. Elices, A general bilinear fit for the softening curve of concrete, Materials and Structures, 27 (1994), 99-105. Google Scholar

[13]

R. GutkinM. L. LaffanS. T. PinhoP. Robinson and P. T. Curtis, Modelling the R-curve effect and its specimen-dependence, Int. J. of Solids and Structures, 48 (2011), 1767-1777. Google Scholar

[14]

J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, Advances in Applied Mechanics, 29 (1992), 63-191. Google Scholar

[15]

M. KočvaraA. Mielke and T. Roubíček, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447. doi: 10.1177/1081286505046482. Google Scholar

[16]

J. Lemaitre and R. Desmorat, Engineering Damage Mechanics, Springer, Berlin, 2005.Google Scholar

[17]

V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface, J. Engr. Mater. Technology, 130 (2008), 045501.Google Scholar

[18]

V. Mantič, A. Blázquez, E. Correa and F. París, Analysis of interface cracks with contact in composites by 2D BEM, In Fracture and Damage of Composites, M. Guagliano and M. H. Aliabadi (Eds. ), pp. 189-248. WIT Press, Southampton, 2006.Google Scholar

[19]

V. Mantič and F. París, Relation between {SIF and ERR based measures of fracture mode mixity in interface cracks, Int. J. Fracture, 130 (2004), 557-569. Google Scholar

[20]

V. MantičL. TávaraA. BlázquezE. Graciani and F. París, A linear elastic-brittle interface model: Application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads, Int. J. Fracture, 195 (2015), 15-38. Google Scholar

[21]

D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin, 2000.Google Scholar

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A. Mielke, Differential, energetic and metric formulations for rate-independent processes Nonlinear PDEs and Applications, 87–170, Lecture Notes in Math. , 2028, C. I. M. E. Summer Sch. , Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21861-3_3. Google Scholar

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A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Applications, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[24]

M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis, Int. J. Num. Meth. Engrg., 44 (1999), 1267-1283. Google Scholar

[25]

C. G. PanagiotopoulosV. Mantič and T. Roubíček, A simple and efficient BEM implementation of quasistatic linear visco-elasticity, Int. J. Solid Struct., 51 (2014), 2261-2271. Google Scholar

[26]

K. Park and G. H. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces, Applied Mechanics Reviews, 64 (2011).Google Scholar

[27]

P. E. Petersson, Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials Crack Growth and Development of Fracture Zones in Plain Concrete, Report TVBM-1006, Lund Institute of Technology, Lund, 1981.Google Scholar

[28]

N. PircF. SchmidtM. MongeauF. Bugarin and F. Chinesta, Optimization of BEM-based cooling channels injection moulding using model reduction, International Journal of Material Forming, 1 (2008), 1043-1046. Google Scholar

[29]

M. RaousL. Cangemi and M. Cocu, A consistent model coupling adhesion, friction and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399. doi: 10.1016/S0045-7825(98)00389-2. Google Scholar

[30]

T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126. doi: 10.1137/12088286X. Google Scholar

[31]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.Google Scholar

[32]

T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their mathematical analysis and numerical treatment, In Mathematical Methods and Models in Composites, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 349{400. doi: 10.1142/9781848167858_0009. Google Scholar

[33]

T. RoubíčekC. Panagiotopoulos and V. Mantič, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840. doi: 10.1002/zamm.201200239. Google Scholar

[34]

A. Sutradhar, G. H. Paulino and L. J. Gray, The Symmetric Galerkin Boundary Element Method, Springer-Verlag, Berlin, 2008. Google Scholar

[35]

L. TávaraV. MantičE. Graciani and F. París, BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model, Eng. Anal. Bound. Elem., 35 (2011), 207-222. doi: 10.1016/j.enganabound.2010.08.006. Google Scholar

[36]

L. TávaraV. MantičA. SalvadoriL. J. Gray and F. París, Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids, Computational Mechanics, 51 (2013), 535-551. doi: 10.1007/s00466-012-0808-5. Google Scholar

[37]

V. Tvergaard and J. W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids, 40 (1992), 1377-1397. doi: 10.1016/0022-5096(92)90020-3. Google Scholar

[38]

T. VandellosC. Huchette and N. Carrere, Proposition of a framework for the development of a cohesive zone model adapted to carbon-fiber reinforced plastic laminated composites, Composite Structures, 105 (2013), 199-206. Google Scholar

[39]

A. Visintin, Models of Phase Transition, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5. Google Scholar

[40]

R. Vodička, A quasi-static interface damage model with cohesive cracks: SQP-SGBEM implementation, Eng. Anal. Bound. Elem., 62 (2016), 123-140. doi: 10.1016/j.enganabound.2015.09.010. Google Scholar

[41]

R. VodičkaV. Mantič and F. París, Symmetric variational formulation of BIE for domain decomposition problems in elasticity -an SGBEM approach for nonconforming discretizations of curved interfaces, CMES -Comp. Model. Eng., 17 (2007), 173-203. Google Scholar

[42]

R. VodičkaV. Mantič and F. París, Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form, Eng. Anal. Bound. Elem., 35 (2011), 148-155. doi: 10.1016/j.enganabound.2010.05.002. Google Scholar

[43]

R. VodičkaV. Mantič and T. Roubíček, Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963. doi: 10.1007/s11012-014-0045-4. Google Scholar

[44]

R. VodičkaV. Mantič and T. Roubíček, Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM, J. Comp. Appl. Math., 315 (2017), 249-272. doi: 10.1016/j.cam.2016.10.010. Google Scholar

[45]

R. Vodička, T. Roubíček and V. Mantič, General-purpose model for various adhesive frictional contacts at small strains, Interfaces Free Bound., (submitted).Google Scholar

[46]

P. Wriggers, Computational Contact Mechanics, Springer, Berlin, 2006.Google Scholar

show all references

References:
[1]

L. Banks-Sills and D. Askenazi, A note on fracture criteria for interface fracture, Int. J. Fracture, 103 (2000), 177-188. Google Scholar

[2]

O. Barani, M. Mosallanejad and S. Sadrnejad, Fracture analysis of cohesive soils using bilinear and trilinear cohesive laws, Int. J. Geomech., 04015088,2015.Google Scholar

[3]

Z. P. Bažant and J. Planas, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, 1998.Google Scholar

[4]

J. Besson, G. Cailletaud, J. L. Chaboche and S. Forest, Non-Linear Mechanics of Materials, Springer, Dordrecht, 2010.Google Scholar

[5]

W. Brocks, A. Cornec and I. Scheider, Computational aspects of Nonlinear Fracture Mechanics (Chapter 3. 03), In: Numerical and Computational Methods (Vol. 3), R. de Borst, H. A. Mang (Vol. Eds. ), Comprehensive Structural: Fracture of Materials from Nano to Macro, I. Milne, R. O. Ritchie, B. Karihaloo (Eds. ), pp. 127{209, Elsevier, 2003.Google Scholar

[6]

A. Carpinteri, Post-peak and post-bifurcation analysis of catastrophic softening behaviour (snap-back instability), Engineering Fracture Mechanics, 32 (1989), 265-278. Google Scholar

[7]

C. G. DávilaC. A. Rose and P. P. Camanho, A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture, International Journal of Fracture, 158 (2009), 211-223. Google Scholar

[8]

C. G. Dávila, C. A. Rose and E. V. Iarve, Modeling fracture and complex crack networks in laminated composites, in Mathematical Methods and Models in Composites, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 297-347. doi: 10.1142/9781848167858_0008. Google Scholar

[9]

G. Del Piero and M. Raous, A unified model for adhesive interfaces with damage, viscosity, and friction, Europ. J. of Mechanics A/Solids, 29 (2010), 496-507. doi: 10.1016/j.euromechsol.2010.02.004. Google Scholar

[10]

Z. Dostál, Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications, Springer, Berlin, 2009. Google Scholar

[11]

M. Frémond, Dissipation dans l'adherence des solides, C.R. Acad. Sci., Paris, Sér.Ⅱ, 300 (1985), 709-714. Google Scholar

[12]

G. V. GuineaJ. Planas and M. Elices, A general bilinear fit for the softening curve of concrete, Materials and Structures, 27 (1994), 99-105. Google Scholar

[13]

R. GutkinM. L. LaffanS. T. PinhoP. Robinson and P. T. Curtis, Modelling the R-curve effect and its specimen-dependence, Int. J. of Solids and Structures, 48 (2011), 1767-1777. Google Scholar

[14]

J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, Advances in Applied Mechanics, 29 (1992), 63-191. Google Scholar

[15]

M. KočvaraA. Mielke and T. Roubíček, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447. doi: 10.1177/1081286505046482. Google Scholar

[16]

J. Lemaitre and R. Desmorat, Engineering Damage Mechanics, Springer, Berlin, 2005.Google Scholar

[17]

V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface, J. Engr. Mater. Technology, 130 (2008), 045501.Google Scholar

[18]

V. Mantič, A. Blázquez, E. Correa and F. París, Analysis of interface cracks with contact in composites by 2D BEM, In Fracture and Damage of Composites, M. Guagliano and M. H. Aliabadi (Eds. ), pp. 189-248. WIT Press, Southampton, 2006.Google Scholar

[19]

V. Mantič and F. París, Relation between {SIF and ERR based measures of fracture mode mixity in interface cracks, Int. J. Fracture, 130 (2004), 557-569. Google Scholar

[20]

V. MantičL. TávaraA. BlázquezE. Graciani and F. París, A linear elastic-brittle interface model: Application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads, Int. J. Fracture, 195 (2015), 15-38. Google Scholar

[21]

D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin, 2000.Google Scholar

[22]

A. Mielke, Differential, energetic and metric formulations for rate-independent processes Nonlinear PDEs and Applications, 87–170, Lecture Notes in Math. , 2028, C. I. M. E. Summer Sch. , Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21861-3_3. Google Scholar

[23]

A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Applications, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[24]

M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis, Int. J. Num. Meth. Engrg., 44 (1999), 1267-1283. Google Scholar

[25]

C. G. PanagiotopoulosV. Mantič and T. Roubíček, A simple and efficient BEM implementation of quasistatic linear visco-elasticity, Int. J. Solid Struct., 51 (2014), 2261-2271. Google Scholar

[26]

K. Park and G. H. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces, Applied Mechanics Reviews, 64 (2011).Google Scholar

[27]

P. E. Petersson, Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials Crack Growth and Development of Fracture Zones in Plain Concrete, Report TVBM-1006, Lund Institute of Technology, Lund, 1981.Google Scholar

[28]

N. PircF. SchmidtM. MongeauF. Bugarin and F. Chinesta, Optimization of BEM-based cooling channels injection moulding using model reduction, International Journal of Material Forming, 1 (2008), 1043-1046. Google Scholar

[29]

M. RaousL. Cangemi and M. Cocu, A consistent model coupling adhesion, friction and unilateral contact, Comput. Methods Appl. Mech. Engrg., 177 (1999), 383-399. doi: 10.1016/S0045-7825(98)00389-2. Google Scholar

[30]

T. Roubíček, Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM J. Math. Anal., 45 (2013), 101-126. doi: 10.1137/12088286X. Google Scholar

[31]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.Google Scholar

[32]

T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their mathematical analysis and numerical treatment, In Mathematical Methods and Models in Composites, V. Mantič (Ed. ), Imperial College Press, 5 (2014), 349{400. doi: 10.1142/9781848167858_0009. Google Scholar

[33]

T. RoubíčekC. Panagiotopoulos and V. Mantič, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840. doi: 10.1002/zamm.201200239. Google Scholar

[34]

A. Sutradhar, G. H. Paulino and L. J. Gray, The Symmetric Galerkin Boundary Element Method, Springer-Verlag, Berlin, 2008. Google Scholar

[35]

L. TávaraV. MantičE. Graciani and F. París, BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model, Eng. Anal. Bound. Elem., 35 (2011), 207-222. doi: 10.1016/j.enganabound.2010.08.006. Google Scholar

[36]

L. TávaraV. MantičA. SalvadoriL. J. Gray and F. París, Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids, Computational Mechanics, 51 (2013), 535-551. doi: 10.1007/s00466-012-0808-5. Google Scholar

[37]

V. Tvergaard and J. W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids, 40 (1992), 1377-1397. doi: 10.1016/0022-5096(92)90020-3. Google Scholar

[38]

T. VandellosC. Huchette and N. Carrere, Proposition of a framework for the development of a cohesive zone model adapted to carbon-fiber reinforced plastic laminated composites, Composite Structures, 105 (2013), 199-206. Google Scholar

[39]

A. Visintin, Models of Phase Transition, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5. Google Scholar

[40]

R. Vodička, A quasi-static interface damage model with cohesive cracks: SQP-SGBEM implementation, Eng. Anal. Bound. Elem., 62 (2016), 123-140. doi: 10.1016/j.enganabound.2015.09.010. Google Scholar

[41]

R. VodičkaV. Mantič and F. París, Symmetric variational formulation of BIE for domain decomposition problems in elasticity -an SGBEM approach for nonconforming discretizations of curved interfaces, CMES -Comp. Model. Eng., 17 (2007), 173-203. Google Scholar

[42]

R. VodičkaV. Mantič and F. París, Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form, Eng. Anal. Bound. Elem., 35 (2011), 148-155. doi: 10.1016/j.enganabound.2010.05.002. Google Scholar

[43]

R. VodičkaV. Mantič and T. Roubíček, Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963. doi: 10.1007/s11012-014-0045-4. Google Scholar

[44]

R. VodičkaV. Mantič and T. Roubíček, Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM, J. Comp. Appl. Math., 315 (2017), 249-272. doi: 10.1016/j.cam.2016.10.010. Google Scholar

[45]

R. Vodička, T. Roubíček and V. Mantič, General-purpose model for various adhesive frictional contacts at small strains, Interfaces Free Bound., (submitted).Google Scholar

[46]

P. Wriggers, Computational Contact Mechanics, Springer, Berlin, 2006.Google Scholar

Figure 1.  The used notation for two bonded domains.
Figure 2.  Stress-displacement curves for (a) the bilinear and (b) multilinear CZMs with $m_{\rm d}{=}3$.
Figure 3.  Stress-displacement relations for multilinear CZMs with $m_{\rm d}{=}3$: (a) Mode 'subsequent', (b) Mode 'at once'.
Figure 4.  An example of stress-relative displacement relation for a multilinear CZM with $m_{\rm d}{=}3$, $\sigma_{\max\,{\rm n}}{=}1.25\sigma_{\max\,{\rm t}}$, $G_{\tiny\rm IIc}{=}2G_{\tiny\rm Ic}$.
Figure 5.  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).
Figure 6.  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'.
Figure 7.  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.
Figure 8.  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.
Figure 9.  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.
Figure 10.  The applied forces for DCB calculated at the place where the vertical displacement is imposed.
Figure 11.  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: $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$).
Figure 12.  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.
Figure 13.  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.
Figure 14.  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.
Figure 15.  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 $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$.
Figure 16.  Stress-relative displacement graphs at selected points of the interface: ${\tau_{\rm r}}{=}0$, $G_{\tiny\rm IIc}{=}4G_{\tiny\rm Ic}$.
Figure 17.  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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