**3. The numerical tests on the three proposed approaches**

The three different mathematical models presented in Section 2 have been applied in two *TC* computation tests. Both tests have been carried out in the Alpine area.

In the first test, the SRTM3 DTM (see [14]) have been selected in the area (called AREA\_1):

$$4\mathfrak{G}^{\circ} \le \mathfrak{q} \le 4\mathcal{T}^{\circ} \qquad \mathbf{11}^{\circ} \le \lambda \le \mathbf{12}^{\circ}$$

The statistics of the height data in this area (see **Figure 6**) are given in **Table 1**. Points for the *TC* computation have been selected on a 30 � 30 regular grid in the inner area:

$$4\text{€}.25^{\circ} \le \mathfrak{q} \le 4\text{€}.75^{\circ} \qquad \text{11.25}^{\circ} \le \lambda \le \text{11.75}^{\circ}$$

The computation points are thus in an inner area which have 0.25° from the extents of the outer one containing the DTM. So, the terrain correction in points

**Figure 6.** *The DTM (AREA\_1) and the points for TC computation.*

*H φ j* þ *Δφ j*, *λ j*,*rj*

DTM point *Q j*

built.

**Figure 5.**

*point* Pi *.*

**50**

. Note that the two planar surfaces identified by the closed poly-

.

. **Figure 5** illustrates in graphical form how the single polyhedron is

gons *ABCD* and *EFGH* have their vertices whose radiuses depend in the first case on the radius of the computation point *Pi* while in the second case on the radius of the

*Geodetic Sciences - Theory, Applications and Recent Developments*

This procedure defines the spherical coordinates of the vertices of the polyhedron that are at the height of the terrain and at the height of the computation point, i.e. the level at which the Bouguer plate is computed. The computation of the firstorder derivative along the direction of *ri* of the gravitational effect of such polyhedron corresponds to the terrain correction to be applied at *Pi* as contribute of *Q j*

Such value is obtained by running the code *polyhedron.f* made available by the author [4] implementing the linear integral approach. As input, the relative cartesian coordinates of the polyhedron vertices with respect to the computation point and their topological relationships are required. These are obtained applying a change of reference system to the polyhedron vertices. In particular, their coordinates were roto-translated into a local reference system having the origin at the computation point *Pi*, the *z* axis pointing up along the direction of *ri* and the *x* and *y* axes parallel to the local East and North directions, respectively. Regarding the topological relationships defining the outer normal direction of the six planes of the polyhedron, they are defined by a topology matrix containing the counterclockwise sequence of the vertices as seen from outside. As output, the absolute value of the computed *Vr* is taken. This procedure contemplates two nested loops over all the *m* DTM points *Q j* and the *n* computation points *Pi*. Within the loop over the computation points, different local reference systems are defined. This leads to slight changes in the directions of the *x* and *y* axes but not on the *z* axis, always normal and pointing outside the reference sphere defined on *Pi*, then maintaining

*Sketch of the polyhedron vertices building procedure on the basis of the DTM point* Qj *and the computation*

mGal. Even though this value is high if compared with the precision of the gravity observations (which can reach few μGals), one has to consider that other error sources in the topography reduction process can have a larger impact. As an example, the discrepancy between the heights of the point associated with the gravity observations as compared with those obtained by the DTM in the same points can amount to ten meters (or even more) in mountain areas. Given that the absolute value of the free-air gradient is 0*:*30877 mGal*=*m, this implies 3 mGal in 10 m due to this mismatch. Also, biases can occur due to the assumption of constant density. In view of that, even the maximum difference between the GRAVSOFT terrain correction and the spherical tesseroid/flat tesseroid values are not so significant.

*The Gravity Effect of Topography: A Comparison among Three Different Methods*

*DOI: http://dx.doi.org/10.5772/intechopen.97718*

A second test was then devised. Observed gravity data were selected in the area

46° ≤φ≤47° 11° ≤λ≤12°

Gravity point coordinates were surveyed with GNSS and framed to ITRF94. Statistics of the ellipsoidal heights of these gravity points are listed in **Table 4**. Gravity values have been measured with a Lacoste&Romberg G-367 relative gravimeter. The standard deviation of the observed values is of the order of 0.02 mGal. Gravity data are referred to IGSN71 and their statistics are summarized in

For the computation of the terrain component, the SRTM3 DTM have been selected in the 3° 3° area centered on the one containing the gravity data area

45° ≤φ≤48° 10°≤λ≤ 13°

**Figure 7** shows the DTM features of AREA\_2 and the position of the gravity

**Number of points μ [m] σ [m] Min [m] Max [m]** 116 1161.74 384.22 312.67 2217.08

**Number of points μ [mGal] σ [mGal] Min [mGal] Max [mGal]** 116 980404.459 71.605 980208.060 980545.806

**Number of points μ [m] σ [m] Min [m] Max [m]** 12967201 1054.4 853.2 35.0 3865.0

Similarly to what has been done in the first test, SRTM3 and gravity point coordinates were transformed into spherical coordinates for the computation of the terrain correction with the UNIPOL and FT approaches. On the other hands, ellipsoidal heights of gravity points have been converted into orthometric heights via

The statistics of the SRTM3 in AREA\_2 are described in **Table 6**.

*The statistics of the heights of the computation points in AREA\_2.*

*The statistics of the observed gravity values.*

*The statistics of the DTM data in AREA\_2.*

(AREA\_2):

**Table 5**.

(AREA\_2)

points.

**Table 4.**

**Table 5.**

**Table 6.**

**53**


*3.3. Strain Hardening Behaviors of CR Samples*

The 10% CR and 30% CR samples with good ductility were selected to study the strain hardening behaviors and their strain hardening rates (SHRs; dσT/dεT) were plotted as a function of true strain, presented in Figure 7. The SHR plots of selected samples can be divided into three stages, as shown in Table 4. The SHRs first decreased (Stage A), then increased (Stage B), before finally decreasing again (Stage C) with increasing true strain.

**Figure 7.** Strain hardening rate for 10% CR (A1, B1, C1) and 30% CR samples (A2, B2, C2).

**Table 4.** Values of plastic strain of the samples at each stage. The data are taken from Figure 7.

#### **4. Discussion**

In this paper, the effect of cold deformation on the mechanical properties of 316LN ASS was studied. It is well-known that the microstructural characteristics are a major factor determining the mechanical properties of steels. Therefore, the effect of cold deformation on microstructural evolution and the relationship between the microstructures and mechanical properties of CR 316LN ASS were discussed. The ductility of the ASSs is related to their strain hardening behavior. To obtain high strength 316LN ASS with a reasonable ductility, the strain hardening behaviors of CR samples were discussed.

#### *4.1. The Effect of Cold Deformation on the Microstructures*

The deformation microstructures formed in ASSs during the plastic deformation process depend on their deformation mechanisms, which are determined by their SFEs. The SFE of the 316LN ASS at room temperature is ~18.9 mJ/m2 (Table 1) and both the strain induced martensite and the mechanical twin could form during plastic deformation [13,20]. When the cold rolling reduction increased, the shear bands comprised of slip bands and mechanical twins formed in austenite grains and then the strain-induced martensite nucleated at the intersection of shear bands [21–23]. The formed ά-martensite replaced the shear bands and the shear bands gradually disappeared as the cold rolling reduction increased. Furthermore, the martensite lath could be broken and mixed with untransformed austenite to form dislocation-cell-type martensite that has a higher density of dislocation (Figure 3f) [24].

Previous studies [25,26] have indicated that heavily cold worked metals could be subdivided by grain boundaries and dislocation boundaries. The cold work inducing the formation of the grain boundary was also found in CR AZ91 alloy [27] and 308L stainless steel [28]. In this 316LN ASS, in addition to dislocation grain boundaries, the boundaries of mechanical twins and strain-induced martensite formed during the CR process can also subdivide untransformed austenite, leading to the decrease of the untransformed austenite structure size. The EBSD results in Figure 4 revealed a similar tendency, where the increased density of grain boundaries led to a decreasing untransformed austenite structure size. Hughes and Hansen's study [29] indicated that the continued subdivision of

grains into crystallites surrounded by dislocation boundaries leads to a large orientation spread based on dislocation accumulation processes. The main grain boundaries formed in the 10% CR sample were LAGB that may originate from the dislocation boundary formed by dislocation accumulated at twins' boundaries or shear bands. The rapidly increasing density of LAGB as the cold reduction increased from 10% to 30% could be attributed to the increased dislocation boundaries caused by the increasing density of dislocation. The difficult dislocation boundaries formed could show an operation of different slip system combinations within the individual crystallites, leading to the possibility of different parts of a grain rotating towards different stable end orientations [29]. If such end orientations are not far apart, large misorientations within the original grain can build up. Considering that the density of the dislocation boundary in the untransformed austenite grains increases with the increasing cold rolling reduction, the probability of large misorientations within the original grains increases with the increasing cold rolling reduction, leading to the increasing density of HAGB. The results in Figures 4 and 5 illustrate this viewpoint very well as the variations of HAGB densities are in good agreement with the variations of the average numbers of misorientations in the original austenite grains.