9.3.2 Cracking

Next: 9.3.3 Bond-slip Up: 9.3 Interface Behavior Previous: 9.3.1 Elasticity Contents Index

Subsections

9.3.2 Cracking

DIANA offers two models to simulate cracking with interface elements: a discrete cracking model and a crack dilatancy model. The discrete cracking model can be used in combination with maturity dependency of the tensile strength f_t .

9.3.2.1 Discrete Cracking

Discrete cracking is specified as initiation, Mode-I behavior and Mode-II behavior. With the models of this section, Mode-I and Mode-II are uncoupled. If coupling is required, use the asymmetric crack dilatancy models as described in the next section. See §21.1 for background theory.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...\>\texttt{MO2VAL}\>\texttt{\textit{mv2}}$_{r}\,${]} \end{tabbing} \end{figure}$

DISCRA: 1 indicates a discrete crack initiation criterion of normal traction. A discrete crack arises if the normal traction t_n exceeds f_t .
DCRVAL: ft is the tensile strength f_t .
TEMCRK: specifies temperature influence of the tensile strength. fta ...ftz are the tensile strengths f_t , (z $\leq$ 30 )respectively valid for the corresponding tea ...tez temperatures.
CONCRK: specifies concentration influence of the tensile strength. fta ...ftz are the tensile strengths f_t , (z $\leq$ 30 )respectively valid for the corresponding coa ...coz concentrations.
MATCRK: specifies maturity influence of the tensile strength. fta ...ftz are the tensile strengths f_t , (z $\leq$ 30 )respectively valid for the corresponding mva ...mvz maturity variables. Equivalent age is the only maturity variable that can be used for this model.
MODE1: mo1 indicates the Mode-I tension softening criterion [Fig.9.10].
MO1VAL: mv1 are values for the Mode-I tension softening criterion.
TEMGF1: specifies temperature influence of the fracture energy. gfa ...gfz are the fracture energies G_f , (z $\leq$ 30 )respectively valid for the corresponding tea ...tez temperatures.
CONGF1: specifies concentration influence of the fracture energy. gfa ...gfz are the fracture energies G_f , (z $\leq$ 30 )respectively valid for the corresponding coa ...coz concentrations.
MATGF1: specifies maturity influence of the fracture energy. gfa ...gfz are the fracture energies G_f , (z $\leq$ 30 )respectively valid for the corresponding mva ...mvz maturity variables. Equivalent age is the only maturity variable that can be used for this model.
UNLO1: un1 specifies a Mode-I unloading/reloading model.
UNLO1 1 for secant unloading: a straight line back to the origin. Beyond the origin, in the compressive regime, return to the linear elastic stiffness. This is the default model. [UNLO1 1]
UNLO1 2 for elastic unloading: immediate return to the linear elastic stiffness.
UNLO1 3 for cyclic unloading via hysteresis loops according to the continuous function model by Hordijk [Fig.21.6]. This model is only applicable in combination with the nonlinear tension softening criterion of Hordijk et al. [Fig.9.10c].
MODE2: mo2 specifies a shear criterion to be used in the crack development stage.
MODE2 0 for zero shear traction and zero shear stiffness after cracking. This is the default model. [MODE2 0]
MODE2 1 for a constant shear modulus after cracking.
MO2VAL: mv2 is the value of the shear modulus to be used in the development stage of the crack. This value is only applied in case of constant shear modulus after cracking.

Tension Softening.

Figure 9.10 shows the available tension softening models.

**Figure 9.10:** Mode-I tension softening - discrete cracks
$\begin{figure}\begin{footnotesize}\setlength{\unitlength}{1cm} \begin{picture... ...enterline{\raise 2.2cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

Brittle cracking (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...extsf{\tiny {80}}}\\ * \>\>\texttt{MODE1}\>\texttt{0} \end{tabbing} \end{figure}$

MODE1: 0 indicates brittle cracking [§21.1.1]. No further values are necessary.

Linear tension softening (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ... \>\>\texttt{MO1VAL}\>\texttt{\textit{gf}}$_{r}\,$ \end{tabbing} \end{figure}$

MODE1: 1 indicates linear tension softening [§21.1.2].
MO1VAL: gf is the fracture energy G_f , defining the area below the diagram.

(file.dat)

'MATERI'
    1  DSTIF   1000. 1000.
       DISCRA  1
       DCRVAL  3.
       MODE1   1
       MO1VAL  0.05
       MODE2   1
       MO2VAL  0.

This example specifies linear tension softening with f_t = 3 N/mm² and G_f = 0.05 N/mm . The corresponding ultimate crack width u_ult is 2G_f/f_t = 0.0333 mm . The shear stiffness is reduced to zero after cracking.

Nonlinear tension softening (Hordijk at al.) (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ... \>\>\texttt{MO1VAL}\>\texttt{\textit{gf}}$_{r}\,$ \end{tabbing} \end{figure}$

MODE1

2 indicates nonlinear tension softening according to Hordijk et al.:

$\begin{displaymath}\begin{split}\frac{t_{n}}{f_{\mathrm{t}}} = \left( 1 + {\left... ...^{0}} \left( 1 + c_{1}^{3} \right) \exp ( - c_{2} ) \end{split}\end{displaymath}$

(9.1)

with c₁ = 3 and c₂ = 6.93 . See also §21.1.3 for background theory.

MO1VAL

gf is the fracture energy G_f , defining the area below the diagram.

(file.dat)

'MATERI'
    2  DSTIF   1000. 1000.
       DISCRA  1
       DCRVAL  3.
       MODE1   2
       MO1VAL  0.05
       MODE2   0

This example specifies nonlinear tension softening with f_t = 3 N/mm² and G_f = 0.05 N/mm . The shear stiffness and shear traction are reduced to zero after cracking.

Multilinear tension softening (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...extit{tnn}}$_{r}\,$ \texttt{\textit{unn}}$_{r}\,$ \end{tabbing} \end{figure}$

MODE1: 3 indicates multilinear tension softening.
MO1VAL: specifies a diagram with two values for each point: tn1 to tnn (n $\leq$ 25 )are the tensile tractions t_n normal to the interface, un1 to unn are the relative displacements $\Delta$ u_n normal to the interface. The diagram may also contain ascending parts (hardening).

(file.dat)

'MATERI'
    3  DSTIF   1000. 1000.
       DISCRA  1
       DCRVAL  3.
       MODE1   3
       MO1VAL  3.0 0.0  1.0 0.0133  0. 0.06
       UNLO1   2
       MODE2   1
       MO2VAL  10.

This example specifies a bilinear tension softening diagram (Hillerborg) with f_t = 3 N/mm² and G_f = 0.05 N/mm . Breakpoints are at ${\tfrac{{1}}{{3}}}$ f_t and ${\tfrac{{2}}{{9}}}$ u_ult . There is elastic unloading. The shear stiffness is reduced by a factor of 100 after cracking.

9.3.2.2 Crack Dilatancy

This section describes the input syntax of crack dilatancy in interface elements. Compared to standard discrete cracking [§9.3.2.1], crack dilatancy applies to more advanced analysis of sliding along rough macro-cracks. Crack dilatancy can be coupled with tension softening, in order to describe the development stage of the macro-crack as well.

Crack dilatancy is only available for two-dimensional interface elements [Table 9.1].

See §21.2 for background theory.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...\>\texttt{MO2VAL}\>\texttt{\textit{mv2}}$_{r}\,${]} \end{tabbing} \end{figure}$

CRDILA: dilnr is the number of the crack dilatancy model.
CRDILA 1 for the contact density model by Li et al. [§21.2.5].
CRDILA 2 for the two-phase model by Walraven [§21.2.4].
CRDILA 3 for the rough crack model by Bazant and Gambarova [§21.2.1].
CRDILA 4 for the aggregate interlock relation by Walraven and Reinhardt [§21.2.3].
CRDILA 5 for the rough crack model by Gambarova and Karakoç [§21.2.2].
DILVAL: describes the crack dilatancy model. Value fcc is the cube compressive strength f_cc . Value ft is the tensile strength f_t . Value dmax is the maximum aggregate size.
MODE1: mo1 is the number of the tension softening criterion in the development stage of the crack.
MODE1 0 for immediate stress drop to zero, i.e., brittle cracking [§18.1.1.1] (the default). [MODE1 0]
MODE1 1 for linear tension softening [§18.1.1.2].
MO1VAL: mv1 describes the tension softening criterion. For brittle cracking mv1 is the initial crack width, beyond this value the dilatancy model is activated. For linear softening mv1 is the fracture energy G_f defining the area below the diagram, beyond the softening diagram the dilatancy model is activated.
MODE2: mo2 is the number of the shear criterion in the development stage of the crack.
MODE2 0 for zero shear traction and zero shear stiffness after cracking (the default). [MODE2 0]
MODE2 1 for constant shear modulus after cracking.
MO2VAL: mv2 is the value of the shear modulus in the development stage of the crack, only necessary for the constant shear modulus criterion.

Two-phase (file.dat)

'MATERI'
  1  DSTIF   1000. 1000.
     CRDILA  2
     DILVAL  35.  3.0  12.
     MODE1   1
     MO1VAL  0.05
     MODE2   0

This example specifies the two-phase model by Walraven with f_cc = 35 N/mm² , f_t = 3 N/mm² and a maximum aggregate size of 12 mm . The model becomes active beyond the linear softening diagram.

Rough crack (file.dat)

'MATERI'
  2  DSTIF   1000. 1000.
     CRDILA  3
     DILVAL  38.5  0.0  16.
     MODE1   0
     MO1VAL  0.009

This example specifies the rough crack model by Bazant and Gambarova. There is a pre-existing crack, with f_cc = 38.5 N/mm² , f_t = 0 and a maximum aggregate size of 16 mm . The model becomes active beyond an initial crack width of 0.009 mm .

Next: 9.3.3 Bond-slip Up: 9.3 Interface Behavior Previous: 9.3.1 Elasticity Contents Index

DIANA-9.3 User's Manual - Material Library
First ed.

Copyright (c) 2008 by TNO DIANA BV.