needleman_debonding#

Description#

This behavior 1this 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 Needleman model 2Needleman A., “A continuum model for void nucleation by inclusion debonding”, J. of Applied Mechanics, 54 (1987), pp. 525-531. is described through a scalar variable \(\lambda\) which characterizes the relative crack opening:

\[\begin{aligned} \lambda=\sqrt{\left( \frac{\left \langle u_N \right \rangle} {\delta_N} \right)^2 + \left( \frac{ \Vert \vec{u}_T \Vert } {\delta_T} \right)^2 }, \end{aligned}\]

where \(\left\langle x \right\rangle =x\) if \(x>0\) and \(\left\langle x \right\rangle =0\) if \(x \leq 0\). With respect to the interface normal \(\vec{n}\), \(\vec{u}_N = \left( \vec{u} \cdot \vec{n} \right) \vec{n} \equiv u_N \vec{n}\) and \(\vec{u}_T \equiv \vec{u} - \vec{u}_N\) denote, respectively, the normal and shear opening displacements and \(\delta_N\) and \(\delta_T\) the corresponding maximum allowable values of their norms. 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

\[\begin{aligned} T_N = \frac{u_N}{\delta_N} F \left( \lambda_{max} \right), & \vec{T}_T = \alpha \frac{\vec{u}_T}{\delta_T} F \left( \lambda_{max} \right), & F \left( \lambda \right) = \frac{27}{4} \sigma_{max} \left( 1-\lambda \right)^2, \end{aligned}\]

with \(\alpha\) a constant representing the relative magnitude of \(\Vert \vec{T}_T \Vert\) with respect to \(T_N\), and \(\sigma_{max}\) the maximum stress allowable by the element. For the compressive case, where \(u_N<0\), the normal component of the traction is modified to

\[\begin{aligned} T_N = \alpha_c \frac{u_N}{\delta_N} F(0), \end{aligned}\]

with \(\alpha_c\) a penalization factor. In the literature, \(\alpha_c\) usually is at least \(10\alpha\). Fig. 13 illustrates the typical response of the cohesive zone model under specific loads, for the parameters as given in the example.

../../_images/Needleman_tu.svg

Fig. 13 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 \(T(t)\)#

Syntax#

***behavior needleman_debonding \(~\,~\,~\,\) sigmax \(\sigma_{max}\) \(~\,~\,~\,\) deltan \(\delta_n\) \(~\,~\,~\,\) deltat \(\delta_t\) \(~\,~\,~\,\) alpha \(\alpha\) \(~\,~\,~\,\) alphac \(\alpha_c\) \(~\,~\,~\,\) [no_penetration]

If no_penetration is specified, a broken element continues to prevent penetration.

Example#

***behavior needleman_debonding
   sigmax  100.
   deltan    1.e-5
   deltat    1.e-5
   alpha     1.
   alphac    1.e3
***return