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 λ which characterizes the relative crack opening:

λ=(uNδN)2+(uTδT)2,

where x=x if x>0 and x=0 if x0. With respect to the interface normal n, uN=(un)nuNn and uTuuN denote, respectively, the normal and shear opening displacements and δN and δT the corresponding maximum allowable values of their norms. The damage variable λmax, which is the maximum value of λ 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 T, i.e. TN=(Tn)nTNn and TT=TTN, are defined by

TN=uNδNF(λmax),TT=αuTδTF(λmax),F(λ)=274σmax(1λ)2,

with α a constant representing the relative magnitude of TT with respect to TN, and σmax the maximum stress allowable by the element. For the compressive case, where uN<0, the normal component of the traction is modified to

TN=αcuNδNF(0),

with αc a penalization factor. In the literature, αc usually is at least 10α. Fig. 12 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. 12 Example in two dimensions of the evolution of the cohesive traction as a function of the opening displacement, for two different loading cases: uN(t) with uT(t)=0 (thick red curves), and uT(t) with uN(t)=0 (thin green curves). Top left: applied load u(t). Note: for 2t4 s, the applied loading becomes negative (but the response uN remains 0 because of an implicit non-penetration condition). Top right: response T(t). Bottom left: λmax(t). Bottom right: u(t) vs T(t)#

Syntax#

***behavior needleman_debonding     sigmax σmax     deltan δn     deltat δt     alpha α     alphac αc     [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