generalized_debonding

Description

This behavior ^{1 1}this behavior is Z-set specific, and therefore does not apply for Z-mat for other codes is used for the special problem of interface debonding. See the command **create_interface_elements (and similar) in the Z-set user manual on how to insert cohesive elements in the mesh. The Generalized model ^{2 2}Vandellos T., Huchette C. and Carrere N., “Proposition of a framework for the development of a cohesive zone model adapted to Carbon-Fiber Reinforced Plastic laminated composites.”, Comp. Struct. 105 (2013), 199-206. is based on the consideration of the characterization of interfacial properties and permits to use three different shapes for the cohesive law (bi-linear, tri-linear and trapezoidal). This model is described through a scalar variable \(\lambda\), which characterizes the relative crack opening:

\[\begin{aligned} \lambda=\left(1-\frac{\left\langle \delta - \delta^{*} \right\rangle}{\left(\delta - \delta^{*} \right)} \right) \left\langle \delta - \delta_{0} \right\rangle \left(\frac{\delta^{*} - \alpha_{\sigma}\delta_{0}}{\delta \left(\delta^{*} - \delta_{0} \right)} \right)+ \left(\frac{\left\langle \delta - \delta^{*} \right\rangle}{\left(\delta - \delta^{*} \right)} \right) \left(1+\frac{\alpha_{\sigma}\delta_{0} \left\langle \delta_{f} - \delta \right\rangle}{\delta \left(\delta^{*} - \delta_{f} \right)} \right), \end{aligned}\]

where \(\left\langle x \right\rangle =x\) if \(x>0\) and \(\left\langle x \right\rangle =0\) if \(x \leq 0\). The parameter \(\delta\) is the relative displacement defined as

\[\begin{aligned} \delta=\sqrt{\left \langle u_N \right \rangle^2 + \Vert \vec{u}_T \Vert^2}. \end{aligned}\]

With respect to the interface normal \(\vec{n}\), \(\vec{u}_N = u_N \vec{n}\) and \(\vec{u}_T \equiv \vec{u} - \vec{u_N}\) denote, respectively, the normal and shear opening displacements. The parameters \(\delta_0\) and \(\delta_f\) are respectively the relative displacement associated with the interfacial strength \(\tau_{0}\) and the one attained when the energy release rate is equal to the fracture toughness \(G_{C}\). It should be noted that the evolution of \(\tau_{0}\) (resp. \(G_{C}\)), as a function of the mixed-mode ratio, is defined by the stress criterion (resp. the propagation law). Finally, the parameter \(\delta^{*}\) is associated with the stress \(\tau^{*}\) attained at the end of the first part of the damage process (Fig. 16). These both parameters depend on the shape of the cohesive law which is defined using two shape parameters : \(\alpha_{\sigma}\) and \(\alpha_{\delta}\) determined by

\[\begin{aligned} \alpha_{\sigma}=\frac{\tau^{*}}{\tau_{0}}, & \alpha_{\delta}=\frac{\left(\delta^{*}-\delta_{0}\right)}{\left(\delta_{f}-\delta_{0}\right)}. \end{aligned}\]

Fig. 16 Shape of the tri-linear model for a constant mixed-mode ratio in the \((\delta, T)\) plane.

The damage variable \(\lambda_{max}\), which is the maximum value of \(\lambda\) reached up until the current instant, increases from 0 (no damage) to 1 (for a broken element). The normal and shear components of the cohesive traction \(\vec{T}\), i.e. \(\vec{T}_N = \left( \vec{T} \cdot \vec{n} \right) \vec{n} \equiv T_N \vec{n}\) and \(\vec{T}_T = \vec{T} - \vec{T}_N\), are defined by:

\[\begin{aligned} T_N = K u_{N} \left( 1 - \lambda \right), & \vec{T}_T = K \vec{u}_T \left( 1 - \lambda \right), \end{aligned}\]

where \(K\) is the interfacial stiffness. For the compressive case, where \(u_N<0\), the normal component of the traction is modified to

\[\begin{aligned} T_N & = & \alpha_c K u_{N}, \end{aligned}\]

with \(\alpha_c\) a penalization factor. Fig. 17 illustrates the typical response of the cohesive zone model under specific loads, for the parameters as given in the example.

Remarks:

• the parameter \(\alpha_{\delta}\) is not required with the bi-linear shape;

• the parameter \(\alpha_{\sigma}\) has to be defined only with the tri-linear law;

• two stress criteria (puissance and reinforcement) and three propagation laws (puissance, benzeggagh and vandellos) are available;

³Turon A., Camanho P.P., Costa J. and Renart J., “Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strengths and elastic stiffness.”, Comp. Struct. 92 (2010), 1857-1864.

• the option *turon is required for applying the modification proposed by Turon ^{3 3}Turon A., Camanho P.P., Costa J. and Renart J., “Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strengths and elastic stiffness.”, Comp. Struct. 92 (2010), 1857-1864..

Syntax

***behavior generalized_cohesive_zone \(~\,~\,\) *strength \(~\,~\,~\,~\,\) Zt \(Zt\) \(~\,~\,~\,~\,\) Sc \(Sc\) \(~\,~\,\) *toughness \(~\,~\,~\,~\,\) G1c \(G_{IC}\) \(~\,~\,~\,~\,\) G2c \(G_{IIC}\) \(~\,~\,\) *stiffness \(~\,~\,~\,~\,\) K \(K\) \(~\,~\,~\,~\,\) alphac \(\alpha_c\) \(~\,~\,\) *propagation propagation law (puissance or benzeggagh or vandellos) \(~\,~\,~\,~\,\) n \(n\) \(~\,~\,\) *initiation stress criterion (puissance or reinforcement) \(~\,~\,\) *softening shape of the cohesive law (bi_linear or tri_linear or trapezoidal) \(~\,~\,~\,~\,\) alpha_delta \(\alpha_{\delta}\) \(~\,~\,~\,~\,\) alpha_sigma \(\alpha_{\sigma}\) \(~\,\) [ *turon ]

Example

***behavior generalized_cohesive_zone
  *strength
   Zt      100.
   Sc      100.
  *toughness
   G1c       0.0005
   G2c       0.0005
  *stiffness
   K         1.e8
   alphac    1.e3
  *propagation benzeggagh
   n 1.142
  *initiation puissance
  *softening tri_linear
   alpha_delta 0.4
   alpha_sigma 0.8
***return

Fig. 17 Example in two dimensions of the evolution of the cohesive traction as a function of the opening displacement, for two different loading cases: \(u_N(t)\) with \(u_T(t)=0\) (thick red curves), and \(u_T(t)\) with \(u_N(t)=0\) (thin green curves). Top left: applied load \(u(t)\). Note: for \(2 \leq t \leq 4\) s, the applied loading becomes negative (but the response \(u_N\) remains \(0\) because of an implicit non-penetration condition). Top right: response \(T(t)\), Bottom left: \(\lambda_{max}(t)\). Bottom right: \(u(t)\) vs \(T(t)\).