4. Precautions and advice for use#

4.1. Assumptions relating to the material used to calculate the displacements#

The calculation of stress intensity factors by extrapolation of displacement jumps assumes knowing the linear elastic characteristics of the cracked material that was used to solve the mechanical problem (calculation of the displacement field). In practice, the POST_K1_K2_K3 command automatically retrieves the material concept affected on a set of cells adjacent to the crack front (for a meshed crack or for X- FEM). The material should be homogeneous, isotropic and linear elastic.

This material can be constant (using DEFI_MATERIAU/ELAS) or function (using DEFI_MATERIAU/ELAS_FO). The only control variables that can be used to calculate stress intensity factors in the case of a functional material are “ TEMP “ (temperature) and “ NEUT1 “. In addition, these variables must necessarily be assigned (in AFFE_MATERIAU/AFFE_VARC) from nodal fields (cham_no).

In the case of a functional material, the elastic parameters are evaluated at the crack bottom calculation points:

  • for a mesh crack (operand FOND_FISS filled in) these points are the nodes at the bottom of the crack;

  • for an X- FEM crack (FISSURE operand filled in), these points are either those contained in the object. FONDFISS from the fiss_xfem data structure, i.e. those derived from it by using the NB_POINT_FOND keyword.

4.2. Reminder on the different methods used to extrapolate movement jumps [R7.02.08]#

At each node at the bottom of the crack, 3 methods (variants) are used to determine \({K}_{I}\), \({K}_{\mathit{II}}\) (and \({K}_{\mathit{III}}\) in 3D).

  • Method 1: for each node in the interpolation segment, we calculate the jump of the displacement field squared and we divide it by \(r\). Different values of \(\mathrm{K1}\) (resp. \(\mathrm{K2}\), \(\mathrm{K3}\)) are obtained (with one multiplying factor) by extrapolation to \(r=0\) of the line segments thus obtained.

  • Method 2: we plot the jump of the displacement field squared according to \(r\). The approximations to \(\mathrm{K1}\) are (always with one multiplying factor) equal to the slope of the segments connecting the origin to the various points of the curve.

  • Method 3: linear regression.

At each node at the bottom of the crack, each method provides a MAX value and a MIN value. At each node at the bottom of the crack, we therefore have 6 values for \({K}_{I}\), 6 values for, 6 values for \({K}_{\mathit{II}}\) and 6 values for \({K}_{\mathit{III}}\) (for method 3, the values MIN and MAX coincide). These values are noted as follows, with the exponent \(j\) corresponding to the method number:

  • for \({K}_{I}\) (\(I\) mode or opening mode): \({K}_{I}^{j,\mathit{MAX}},{K}_{I}^{j,\mathit{MIN}},j\mathrm{=}\mathrm{1,2}\mathrm{,3}\)

  • for \({K}_{\mathit{II}}\) (\(\mathit{II}\) mode or plane shear): \({K}_{\mathit{II}}^{j,\mathit{MAX}},{K}_{\mathit{II}}^{j,\mathit{MIN}},j\mathrm{=}\mathrm{1,2}\mathrm{,3}\)

  • and for \({K}_{\mathit{III}}\) (\(\mathit{III}\) mode or anti-plane shear): \({K}_{\mathit{III}}^{j,\mathit{MAX}},{K}_{\mathit{III}}^{j,\mathit{MIN}},j\mathrm{=}\mathrm{1,2}\mathrm{,3}\).

4.3. Table produced#

Command POST_K1_K2_K3 produces a table-like concept. The table can be printed by IMPR_TABLE [U4.91.03]. For each node at the bottom of the crack, it contains:

  • the values of the corresponding stress intensity factors: K1, K2 (and K3 if 3D)

  • the value of the energy return rate: \(G\)

  • estimates of the error on the stress intensity factors: ERR_K1, ERR_K2 (and ERR_K3 if 3D).

The following paragraphs detail these quantities.

4.3.1. Stress intensity factor values#

The table produced contains, for each node (or point) at the bottom of the crack, the values of the stress intensity factors corresponding to the MAX values of method 1 (see § 4.2):

\(\mathit{K1}\) (\(\text{}\mathrm{=}{K}_{I}^{\mathrm{1,}\mathit{MAX}}\)), \(\mathit{K2}\) (\(\text{}\mathrm{=}{K}_{\mathit{II}}^{\mathrm{1,}\mathit{MAX}}\))

In 3D, we also have \(\mathit{K3}\) (\(\text{}\mathrm{=}{K}_{\mathit{III}}^{\mathrm{1,}\mathit{MAX}}\))

We therefore print a single value of \(\mathit{K1}\), \(\mathit{K2}\) (and \(\mathit{K3}\) if 3D) per node at the bottom of the crack.

4.3.2. Values of the energy return rate#

The table produced contains, for each node (or point) at the bottom of the crack, the value of the energy restoration rate \(G\) calculated from the stress intensity factors by the Irwin formula.

4.3.3. Estimation of the error committed on stress intensity factors#

In order to assess the error made on the stress intensity factors at each bottom node, the difference between the 6 values given by the 3 methods is evaluated (see § 4.2). This gives an absolute discrepancy regarding \(\mathit{K1}\), \(\mathit{K2}\) (and \(\mathit{K3}\) if 3D). To obtain a relative deviation that is easier to interpret, we normalize the absolute differences by a value \(K\) which is the maximum value of all the \(K\) at this node at the bottom of the crack.

More precisely, the error committed on \({K}_{i}(i\mathrm{=}\mathrm{1,2}\mathrm{,3})\), \(i\) being the mode of stress on the crack and \(j\) the number of the method, is defined as follows:

\(\mathit{erreur}({K}_{i})\mathrm{=}\frac{\underset{j\mathrm{=}\mathrm{1,2}\mathrm{,3}}{\mathit{max}}({K}_{i}^{j,\mathit{MAX}}\mathrm{-}{K}_{i}^{j,\mathit{MIN}})}{K}\), with \(K\mathrm{=}\underset{j\mathrm{=}\mathrm{1,2}\mathrm{,3}}{\mathit{max}}({K}_{I}^{j,\mathit{MAX}},{K}_{\mathit{II}}^{j,\mathit{MAX}},{K}_{\mathit{III}}^{j,\mathit{MAX}})\)

The errors on \(\mathit{K1}\), \(\mathit{K2}\) (and \(\mathit{K3}\) if 3D) are printed in the table: ERR_K1, ERR_K2 (and ERR_K3 if 3d) for each node at the bottom of the crack.

4.4. Additional impressions#

If INFO is 2, all intermediate calculations are shown in the message file. It is noted that the column entitled SAUT_DX (respectively SAUT_DY and SAUT_DZ) in the tables of the message file corresponds to the jump in movement along the \(\mathit{X1}\) axis (\(\mathit{X2}\) and \(\mathit{X3}\) respectively), multiplied by a coefficient depending on the material, all squared [R7.02.08].

4.5. Precautions and advice#

The hypotheses necessary for the validity of this method are:

  1. the crack must be sufficiently regular (i.e. the bottom and the lips do not have any geometric particularity);

  2. the behavior must be elastic, linear, isotropic and homogeneous;

  3. the structure must be isothermal (or, at a minimum, the temperature gradients on the lips may be overlooked in the interpolation zone).

The method used is theoretically less accurate and more sensitive to meshing than the singular displacement method [R7.02.05]. In general, it is advisable to compare the results of POST_K1_K2_K3 and those of CALC_G [U4.82.03] in studies, which is a good indicator of the quality of the result obtained.

Tips in the case of mesh cracks: the mesh should preferably be quadratic and have enough nodes perpendicular to the crack bottom. On the other hand, the results are significantly improved if, in the case where the mesh is composed of quadratic elements, the middle nodes (edges that touch the bottom of the crack) are moved to a quarter of these edges by bringing them closer to the bottom of the crack. This is made possible by the MODI_MAILLE keyword (option “NOEUD_QUART”) in the MODI_MAILLAGE [U4.23.04] command.

The calculation by interpolation of displacement jumps requires having at least 3 knots on the normal at the bottom of the crack. If the number of nodes is not sufficient, an alarm is issued and the rows corresponding to this background node are set to 0 in the result table. The calculation then continues, if necessary, for the next node in the crack bottom. In this case, we can:

  • either increase the maximum curvilinear abscissa ABSC_CURV_MAXI to find nodes farther from the bottom of the crack;

  • or increase the parameter PREC_VIS_A_VIS (and possibly PREC_NORM in DEFI_FOND_FISS), which means being less demanding in selecting nodes for the calculation.

Advice in the case of non-meshed cracks: the precision of the method is sensitive to the choice of the enrichment zone for the X- FEM method (parameter RAYON_ENRI of DEFI_FISS_XFEM). Ideally, the enrichment radius and the maximum curvilinear abscissa ABSC_CURV_MAXI are of the order of three times the size of the minimum mesh edge.

Calculations are possible on a non-planar crack, but the user must ensure that it remains sufficiently regular for the calculation hypotheses to be valid: there should be no geometric singularity on the bottom or on the lips. Typically, the calculation is legal for an axisymmetric crack, but not for a corner.

The calculation by interpolation of displacement jumps requires having at least 3 knots on the normal at the bottom of the crack. The number of interpolation points is normally equal to NB_NOEUD_COUPE but may be less in one case:

    • if the geometry of the background and the structure is such that some of the interpolation points leave the material. In this case, it is necessary to reduce ABSC_CURV_MAXI (while remaining consistent with the fineness of the mesh) and/or increase NB_NOEUD_COUPE.

The calculations are quite time and memory intensive if there are a lot of points on the bottom of the crack. The use of the NB_POINT_FOND keyword makes it possible to limit post-processing to a certain number of points evenly distributed along the background (for example, twenty points are often sufficient).