Ⅳ THREE DIMENSIONAL ELASTIC SOLUTIONS OF A
6.2 E XAMPLE R ESULTS FOR F ULL -S PACE P ROBLEM
Using the elastic properties of the hypothetical rocks listing in Table 6.1, the effect of dip angle (φ) on the displacements and stresses of the position of x=y=z=1 induced by the Pz at origin (0,0,0) are depicted in Figs. 6.2-6.3. In addition, the effect of ϕ on the stresses resulting from Pz at φ=90° and ξ=45° is shown in Fig. 6.4.
Figs. 6.2(a)-6.2(c) show the normalized displacements ux*r/Pz (Fig. 6.2(a)), uy*r/Pz
(Fig. 6.2(b)), and uz*r/Pz (Fig. 6.2(c)) at the position x=y=z=1 vs. the rotation of the transversely isotropic planes (φ), due to a vertical point load (Pz) at the origin, for the constituted isotropic/transversely isotropic rocks (Rock 1/Rocks 2-7, Table 6.1). Fig.
6.2(a) depicts the normalized displacement ux of the rocks, induced by Pz. It is observed that any value in each curve is symmetric with respect to the origin of the co-ordinates, and the ratios E/E′ (Rocks 2 and 3), ν/ν′ (Rocks 4 and 5), and G/G′ (Rocks 6 and 7) all strongly influence this displacement. This figure also exhibits that the
magnitude of the normalized induced displacement (0.00026 m2/GN) for Rock 1 is independent of the change in φ. However, for Rocks 2 and 3, the displacement is maximal at about φ=0°-180°, and is minimal at approximately φ=60°-240°. As for Rocks 6 and 7, the displacement is maximal at around φ=50°-230°, and is minimal at about φ=100°-280°. Fig. 6.2(b) presents the normalized displacement uy of the rocks, due to Pz. This figure clearly reveals that the displacement induced in transversely isotropic rocks is deeply affected by the ratios E/E′ (Rocks 2 and 3) and G/G′ (Rocks 6 and 7), but is only slightly influenced by ν/ν′ (Rocks 4 and 5). Notably, the normalized displacement (0.00026 m2/GN) of the isotropic rock (Rock 1) is also independent of φ. Nevertheless, it is found that the values of induced displacement for Rocks 2 and 3 would be partially within the range of -0.0004 to 0, meaning there could be an opposite-direction displacement occurred in these media. Fig. 6.2(c) displays the normalized displacement uz of the rocks, subjected to Pz. Clearly, the ratios E/E′ (Rocks 2 and 3) and G/G′ (Rocks 6 and 7) profoundly impact the induced displacement, but the effect of ν/ν′ (Rocks 4 and 5) on it is little. The magnitude of the normalized induced displacement for Rock 1 is always 0.00179 m2/GN; however, for Rocks 2, 3, 6, and 7, the values of uz are nearly greater than those of Rock 1. The calculated results for the displacement fields are all in good agreement with Wang and Liao′s solutions (1999) if the full-space is homogeneous, linearly elastic, and the planes of transversely isotropy are parallel to the horizontal axes.
Figs. 6.3(a)-6.3(f) plot the non-dimensional normal stresses σxx*r2/Pz (Fig. 6.3(a)), σyy*r2/Pz (Fig. 6.3(b)), σzz*r2/Pz (Fig. 6.3(c)), and the non-dimensional shear stresses
τyz*r2/Pz (Fig. 6.3(d)), τzx*r2/Pz (Fig. 6.3(e)), τxy*r2/Pz (Fig. 6.3(f)), vs. the rotation of the transversely isotropic planes (φ), subjected to a vertical point load (Pz), at x=y=z=1, for the isotropic (Rock 1) and transversely isotropic rocks (Rocks 2-7). Fig. 6.3(a) illustrates the effect of φ on σxx*r2/Pz, for Rocks 1-7. This figure shows the induced stress for the isotropic rock (Rock 1) has the same value (0.005105), that is again independent of φ. However, it is found that the values of induced stress for Rocks 1-7 varying between -0.004 and 0.02, namely, there is an obvious tensile stress occurred in Rock 7. In addition, any value in each curve is symmetric with respect to the origin of the co-ordinates. Hence, from this figure, it is apparently revealed that the induced stress is greatly influenced by the rotation of the transversely isotropic planes (φ), and the type and degree of rock anisotropy (E/E′, ν/ν′, G/G′). Fig. 6.3(b) presents the effect of φ on σyy*r2/Pz, for Rocks 1-7. Notably, the value in the curves is also symmetric with respect to the origin of the co-ordinates, and the ratios E/E′ (Rocks 2 and 3), ν/ν′ (Rocks 4 and 5), and G/G′ (Rocks 6 and 7) do also have a considerable influence on the stress. This graph exhibits the magnitude of the non-dimensional normal stress (σyy*r2/Pz) for Rock 1 (0.005105) is also independent of φ, and the value of the non-dimensional stress is within 0.06. In particular, the computed results of Rock 4/Rock 5 are totally great/less than those of Rock 1. Fig. 6.3(c) depicts the effect of φ on σzz*r2/Pz, for Rocks 1-7. This stress depends heavily on the ratios E/E′ (Rocks 2 and 3) and G/G′ (Rocks 6 and 7); nevertheless, the effect of the ratios ν/ν′ (Rocks 4 and 5) on it is slight. The maximum value of the non-dimensional stress approaches 0.026. Fig. 6.3(d) plots the effect of φ on τyz*r2/Pz, for Rocks 1-7. Evidently, the
ratios E/E′ (Rocks 2 and 3) and G/G′ (Rocks 6 and 7) could intensely affect the induced stress; however, the effect of the ratios ν/ν′ (Rocks 4 and 5) on it is still little. The trend of these stress curves in this figure is similar to that in Fig. 6.3(c). Fig. 6.3(e) displays the effect of φ on τzx*r2/Pz, for Rocks 1-7. The maximum value of the non-dimensional stress is about 0.026. Fig. 6.3(f) shows the effect of φ on τxy*r2/Pz, for Rocks 1-7. The effect of the ratios ν/ν′ (Rocks 4 and 5) in this figure is more explicit than another shear stresses (Figs. 6.3(d) and 6.3(e)). Especially, the calculated results of Rock 4/Rock 5 are great/less than those of Rock 1. The maximum value of the non-dimensional stress is within the range of 0.024. The computed results for the stress fields are exactly identical with those estimated from Wang and Liao′s solutions (1999), in which the planes of transversely isotropic full-space are parallel to the horizontal loading surface.
Figs. 6.4(a)-6.4(f) plot the non-dimensional normal stresses (σxx*r2/Pz, σyy*r2/Pz, σzz*r2/Pz), and the non-dimensional shear stresses (τyz*r2/Pz, τzx*r2/Pz, τxy*r2/Pz), vs. the geometric position ϕ (from 0° to 360°), due to a vertical point load (Pz), at the rotation of the transversely isotropic planes φ=90° and the geometric position ξ=45°, for the constituted isotropic/transversely isotropic rocks (Rock 1/Rocks 2-7). Fig. 6.4(a) clarifies the effect of ϕ on σxx*r2/Pz, for Rocks 1-7. It is observed that the magnitudes of the estimated stress are symmetric with respect to ϕ=180°. The upper/lower part of this figure denotes the compressive/tensile stress occurred in the rock media. The maximum values of tensile/compressive stress appeared at ϕ=0°/180° in Rock 7. In addition, the induced stresses are found to be influenced by the ratios E/E′ (Rocks 2 and
3), ν/ν′ (Rocks 4 and 5), G/G′ (Rocks 6 and 7), and they are all zero at ϕ=90° and 270°.
Fig. 6.4(b) demonstrates the effect of ϕ on σyy*r2/Pz, for Rocks 1-7. Results reveal that the magnitudes of the computed stress are also symmetric with respect to ϕ=180°, and the tensile and compressive stresses would be occurred in all media. However, the maximum values of tensile/compressive stress approximately appeared at ϕ=125° and 235°/55° and 305° in Rock 4. That means at a given position (φ=90° and ξ=45°), the decrease of the ratio ν/ν′ from 1.0 (Rock 1) to 0.75 (Rock 4) could remarkably affect the stress (σyy). Fig. 6.4(c) shows the induced non-dimensional normal stress σzz*r2/Pz for Rocks 1-7. The distributions and magnitudes of the calculated stress are quite different from those of Figs. 6.4(a) and 6.4(b). The tensile/compressive stress can be found within ϕ=0°-90° and 270°-360°/90°-270°. Moreover, the stress (σzz) is apparently impacted by the ratios G/G′ (Rocks 6 and 7); nevertheless, it is little affected by the ratios E/E′ (Rocks 2 and 3) and υ/υ′ (Rocks 4 and 5). The induced non-dimensional shear stress τyz*r2/Pz for Rocks 1-7 is depicted in Fig. 6.4(d). It is noted that the positive/negative values of τyz are respectively symmetric with respect to ϕ=180°.
Additionally, the computed stresses are all zero at ϕ=0°, 180°, and 360°. The results of Rocks 2, 4, 6, 7 are rather distinct from those of Rocks 1, 3, 5. Similarly, the trends can be discovered in Fig. 6.4(e) for τzx*r2/Pz. Eventually, the induced non-dimensional shear stress τxy*r2/Pz for Rocks 1-7 is displayed in Fig. 6.4(f). The calculated positive/negative values of τxy are symmetric with ϕ=90° and 270°. The zero values for τxy are found at ϕ=0°, 90°, 180°, 270°, and 360°. Furthermore, the influences of the type and degree of rock anisotropy in this figure are more explicit than those in Figs.
6.4(d) and 6.4(e). That signifies again that at φ=90° and ξ=45°, the normal and shear stresses owing to a vertical point load are strongly impacted by the geometric position (ϕ) and rock anisotropy (E/E′, υ/υ′, G/G′).
The examples are presented to illustrate the derived solutions and demonstrate how the rotation of transversely isotropic planes (φ), the geometric position (r, ϕ, ξ), and the degree and type of material anisotropy (E/E′, υ/υ′, G/G′) would influence the normalized displacements and non-dimensional normal and shear stresses. Results reveal that the displacements and stresses in the inclined isotropic/transversely isotropic rocks (Rock 1/Rocks 2-7) due to a vertical point load are quite different from those solutions by assuming the transversely isotropic planes are parallel to the horizontal surface. Hence, it is imperative to consider the dip at an angle of inclination when calculating the induced displacements and stresses in a transversely isotropic material by applied loads.
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ux*r Pz (m2
GN) Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(a)
Fig. 6.2.(a) At the position x=y=z=1, the effect of φ on the normalized displacement ux*r/Pz
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Pz ( m2 GN) Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
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Fig. 6.2.(b) At the position x=y=z=1, the effect of φ on the normalized displacement uy*r/Pz
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uz*r Pz (m2
GN) Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(c)
Fig. 6.2.(c) At the position x=y=z=1, the effect of φ on the normalized displacement uz*r/Pz
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Pz Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
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Fig. 6.3.(a) At the position x=y=z=1, the effect of φ on the non-dimensional normal stress σxx*r2/Pz
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0 0.004 0.008 0.012 0.016 0.02 σyy*r2
Pz Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(b)
Fig. 6.3.(b) At the position x=y=z=1, the effect of φ on the non-dimensional normal stress σyy*r2/Pz
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0.008 0.012 0.016 0.02 0.024 0.028 σzz*r2
Pz Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(c)
Fig. 6.3.(c) At the position x=y=z=1, the effect of φ on the non-dimensional normal stress σzz*r2/Pz
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0.008 0.012 0.016 0.02 0.024 0.028 τyz*r2
Pz Loading is Pz
Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(d)
Fig. 6.3.(d) At the position x=y=z=1, the effect of φ on the non-dimensional shear stress τyz*r2/Pz
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Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
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Fig. 6.3.(e) At the position x=y=z=1, the effect of φ on the non-dimensional shear stress τzx*r2/Pz
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Effect of φ (x=y=z=1)
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(f)
Fig. 6.3.(f) At the position x=y=z=1, the effect of φ on the non-dimensional shear stress τxy*r2/Pz
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σxx*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(a)
Fig. 6.4.(a) At the position φ=90°, ξ=45°, the effect of ϕ on the non-dimensional normal stress σxx*r2/Pz
0 45 90 135 180 225 270 315 360 ϕ (ο)
-0.04 -0.02 0 0.02 0.04
σyy*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(b)
Fig. 6.4.(b) At the position φ=90°, ξ=45°, the effect of ϕ on non-dimensional normal stress σyy*r2/Pz
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σzz*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(c)
Fig. 6.4.(c) At the position φ=90°, ξ=45°, the effect of ϕ on the non-dimensional normal stress σzz*r2/Pz
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τyz*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(d)
Fig. 6.4.(d) At the position φ=90°, ξ=45°, the effect of ϕ on the non-dimensional shear stress τyz*r2/Pz
0 45 90 135 180 225 270 315 360 ϕ (ο)
-0.08 -0.04 0 0.04 0.08
τzx*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(e)
Fig. 6.4.(e) At the position φ=90°, ξ=45°, the effect of ϕ on the non-dimensional shear stress τzx*r2/Pz
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-0.06 -0.04 -0.02 0 0.02 0.04 0.06
τxy*r2 Pz
Loading is Pz Effect of ϕ when φ=90ο, ξ=45ο
Rock 1 Rock 2 Rock 3 Rock 4 Rock 5 Rock 6 Rock 7
(f)
Fig. 6.4.(f) At the position φ=90°, ξ=45°, the effect of ϕ on the non-dimensional shear stress τxy*r2/Pz