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The form of the stress \(\sigma _yy \) at \(y=0\) versus x is equivalent to that of \(\sigma _\theta \theta \) versus r shown in Fig. 1a and is identical to that used by Terada (1976). Other stress functions could be used to generate a residual stress field and would give different detailed results to those presented here. Our aim however is to explore the effect of a simple residual stress field on the behaviour of a crack, although our simple residual stress field includes all the general attributes of a residual stress field.

In this section the elastic behaviour of a crack in a residual stress field is studied when subjected to additional tensile or compressive uniaxial applied stress. The geometry of the crack is shown in Fig. 1b. Depending on the length of the crack, the level of applied stress and the magnitude of the residual stresses the crack may be closed, partially open or fully open. These different crack states are defined in Fig. 5. A crack behaviour map may be used to describe the behaviour of the crack, where one axis represents the length of the crack and the other the level of the applied stress. Different states of opening of the crack occupy different regions of the crack behaviour map. We first present the crack behaviour map for positive \(\sigma _\mathrm RS \), that is when the tangential residual stress is tensile at the centre. Throughout this paper we will refer to this case as tensile residual stress. We then describe the analysis that allows the boundaries between the regions of crack behaviour to be found. Finally, we present the crack behaviour map for negative \(\sigma _\mathrm RS \), referred to here as the case of compressive residual stress.

Figure 6 shows the behaviour of the crack for tensile residual stress and tensile and compressive applied stress. The half-length of the crack c is normalised with respect to the size of the tensile region of the residual stress field R and the applied stress \(\sigma _\mathrm APP \) is normalised with respect to \(\sigma _\mathrm RS \). As an example, the line segment AC in Fig. 6 forms the boundary between two regions of crack behaviour. Above the line, for higher magnitudes of applied stress, the crack is fully open. Below the line the crack is partially open: closed at the tip but open in the centre. These two crack states are defined in Fig. 5.

In addition to the behaviour of the crack for different applied stresses, stress intensity factors may also be calculated. For example, Fig. 7 shows the total stress intensity for a crack of half-length \(c=R\) versus the applied stress. This corresponds to the vertical dashed line in Fig. 6 where \(c/R=1\). At the point marked X the crack begins to open at the centre but is closed at the tip. At the point marked Y, when \(\sigma _\mathrm APP /\sigma _\mathrm RS \approx -0.4446\), the crack is open completely. The stress intensity factor is zero for points between X and Y. For points above Y the total stress intensity factor \(K_\mathrm TOT \) is calculated by

where \(K_\mathrm APP \) and \(K_\mathrm RS \) are given by Eq. (7). Note that when \(\sigma _\mathrm APP =0\), the point Z in Fig. 6, \(K_\mathrm APP =0\) and \(K_\mathrm TOT =K_RS \approx 0.4446\;\sigma _\mathrm RS \sqrt\pi c\). As shown in Fig. 7, once the crack opens at the tip, the total stress intensity varies linearly with the applied stress.

Elastic normalised stress intensity factor \(K/\sigma _RS \sqrt\pi c\) and normalised crack opening a / c versus normalised applied stress \(\sigma _APP /\sigma _RS \) for tensile residual stress and a normalised crack length of \(c/R=1\)

where \(K_\mathrm APP \) and \(K_\mathrm RS \) are given by the same expressions as in Eq. (7), except that c is replaced by a. Figure 7 also shows the half-length of the open portion of the crack a versus the applied stress. The crack begins to open at an applied stress of \(\sigma _\mathrm APP /\sigma _\mathrm RS =-1\) and is open completely (\(a/c=1)\) for applied stresses higher than \(\sigma _\mathrm APP /\sigma _\mathrm RS \approx -0.4446\).

The behaviour of the crack for negative \(\sigma _\mathrm RS \), that is when the tangential residual stress is compressive at the centre, is shown in Fig. 9. The half-length of the crack c is normalised with respect to the size of the compressive region of the residual stress field R and the applied stress \(\sigma _\mathrm APP \) is normalised with respect to \(\left \sigma _\mathrm RS \right \).

Elastic normalised stress intensity factor \(K/\sigma _RS \sqrt\pi c\) and normalised crack opening a / c versus normalised applied stress \(\sigma _APP /\sigma _RS \) for compressive residual stress and a normalised crack length of \(c/R=1\)

In the same way as for tensile residual stress, stress intensity factors may be calculated for cracks in a compressive residual stress field with superimposed applied stress. For example, Fig. 10 shows the stress intensity factor normalised by \(\sigma _\mathrm RS \sqrt\pi c\) for a crack of half-length \(c=R\), versus the applied stress. This corresponds to the vertical dashed line in Fig. 9 with \(c/R=1\). At the point marked X in Fig. 9 the crack begins to open at the tip but is closed at the centre. At the point marked Y the crack is open completely. The applied stress corresponding to point Y is calculated to be given by \(\sigma _\mathrm APP / \approx 0.7910\). The procedure for calculating the stress intensity factor for points between X and Y is first to choose a value for the partially open half-length a (see Fig. 5 for the geometry of a partially open crack, open at the tip and closed at the centre) where \(0

For tensile residual stress we consider an initial state where the applied stress is sufficient to ensure the crack is fully closed. For a crack of half-length \(c=R\) the applied stress \(\sigma _\mathrm APP \) must be less than (more compressive than) \(-\sigma _\mathrm RS \). As the applied stress is increased the crack opens first in the centre, as described in the previous section. It is only when the crack has opened fully to the tip that the small-scale yielding behaviour is different to the elastic behaviour. When the applied stress is larger than that to open the crack fully the strip yield model may be used to calculate the effective stress intensity factor.

Again, we consider an initial state where the applied stress is sufficient to ensure the crack is fully closed. For a crack of half-length \(c=R\) the applied stress \(\sigma _\mathrm APP \) must be less than zero. For the case of a compressive residual stress the crack opens first at the tips and therefore the small-scale yielding behaviour will be different to the elastic behaviour. The strip yield model may still be used to calculate the effective stress intensity factor but complicated by the need to use a twin crack geometry. When the applied stress is high enough to open the crack completely, the same strip yield model as for tensile residual stress case can be used.

Finally, Fig. 15 compares the results of the strip yield model for tensile residual stress and compressive residual stress, where the magnitude of the residual stress is equal to the yield stress, that is \(\sigma _\mathrmY /\left=1\). Figure 15 also shows the results for the standard strip yield model, the case with no residual stress. Finite element results are also shown. The kink in the strip yield model results for compressive residual stress at \(\sigma _\mathrm APP /\left\approx 0.6957\) corresponds to the point when the crack becomes fully open. This compares with the corresponding point for linear elastic behaviour which occurs at \(\sigma _\mathrm APP /\left\approx 0.7910\). Therefore, the effect of plasticity on the crack behaviour map for compressive residual stress of Fig. 9 is to lower the line AB slightly.

Table 1 shows the results of the strip yield model and the finite element study for the case of no residual stress, tensile residual stress and compressive residual stress. Values of the normalised effective stress intensity factor are given for three different crack sizes \(c/R=0.6,\;1.0,\;1.6\) and one magnitude of the yield stress relative to the applied stress defined by \(\sigma _\mathrmY /\sigma _\mathrm APP =1.5\).

For the case of no residual stress, effective stress intensities are higher for longer cracks but changing the plane condition does not have a significant effect, although the plane stress values are slightly higher than plane strain. For the range of conditions considered in the table, the strip yield results are within 3% of the finite element results for plane stress. For the case of tensile residual stress, the magnitude of the residual stress field is defined by \(\sigma _\mathrm RS /\sigma _\mathrm APP =1\). Again, plane conditions do not have a significant effect and the agreement between the strip yield and finite element results is very good.