**5.3 Strength characteristics**

**Figure 15** shows the typical stress-strain relationship of the CD tests on four representative red-brown samples of pure weathered soil under four different confining pressures, where *σ*<sup>1</sup> is the axial stress, *σ*<sup>3</sup> is the confining pressure, and *dε*<sup>1</sup> is the axial strain. **Figure 16** shows this typical *ε*v-*ε*<sup>1</sup> relationship of the CD tests, where *ε*<sup>v</sup> is the volumetric strain.

It can be found in **Figure 15** that the peak deviator stress increases as the confining pressure increases. The internal particles of weathered granite soil could be overturned, stridden, and dislocated under lower confining pressure and could

**Figure 14.** *Change in CBR for soils with differing clay content.*

the results of large-scale triaxial tests, the Mohr's circles and the strength failure envelope can be depicted in **Figure 17** on the basis of the Mohr-Coulomb strength criterion. The linear Mohr's envelope of the weathered granite soil in this work can

Eq. (5) shows that the cohesive strength of the samples is 55 kPa. However, because the weathered granite soil belongs to coarse grained soil, the cohesive strengths of the samples should be close to zero. It has been proved that the particles of coarse grained soil might be broken and its peak strength is nonlinear [19]. **Figure 18** shows that the ratio (*τ*f/*σ*) is not constant but decreases when *σ* increases. In addition, the Mohr envelope is bending downward. The radius of bend reduces as the confining pressure increases, which reflects that the particle breakage is con-

According to De-Mello [20], the power function of weathered granite soil in this

As shown in **Figure 18**, the fitting envelope has a good correlation, which further explains that weathered granite soil has the property of nonlinear strength because of particle breakage. The nonlinear strength characteristics of compacted weathered granite soils could be affected by rock strength, intergranular friction, occlusal effect, etc. The mechanism of the nonlinear strength could be interpreted by means of the variation of failure surface angle in previous literature [21]. Under lower confining pressure, the internal friction angle was relatively large, the bending of the strength envelope was obvious, and the samples had the obvious characteristics of shear dilation and particle breakage. With the increase of confining pressure, the internal friction angle decreased and the bending degree of strength envelope decreased gradually. It was showed that under higher confining pressure, the strength characteristics of compacted weathered granite soils were mainly domi-

where *τ*<sup>f</sup> is the peak shear stress and *σ* is the normal stress.

trolled by the confining pressures noticeably.

study can be expressed as follows:

nated by particle breakage.

**Figure 17.**

**155**

*Mohr strength envelope of red-brown samples.*

*τ*<sup>f</sup> ¼ *σ* tan 41*:*3° þ 0*:*55 MPa*,* (5)

*<sup>τ</sup>*<sup>f</sup> <sup>¼</sup> <sup>0</sup>*:*361*σ*<sup>0</sup>*:*<sup>803</sup> (6)

be expressed as

*Weathered Granite Soils*

*DOI: http://dx.doi.org/10.5772/intechopen.86430*

**Figure 15.** *Stress-strain curves.*

**Figure 16.** *Curves of εv-ε1.*

be crushed, filled, and compacted under higher confining pressure. It was eventually manifested that the specimens transit gradually from strain softening status (e.g., when the initial confining pressure was 100 kPa) to strain hardening status (e.g., when the confining pressure was 200 kPa) with the increase of confining pressures on the samples. At any event, there exists a critical confining pressure. The samples soften in strain when the experiment confining pressures were lower than the critical confining pressures and harden in strain when the experimental confining pressures were higher than the critical confining pressures [18]. Thereafter, it could be deduced that the critical confining pressure in this study was between 100 and 200 kPa. It can be observed from **Figure 16** that the samples transit generally from shear dilation to shear shrinkage, and the volumetric strain of the transition point of the specimens increases gradually with the increase of confining pressures on the samples. In summary, as the confining pressure increases, the volumetric strain at failure changes toward volume contraction. According to

the results of large-scale triaxial tests, the Mohr's circles and the strength failure envelope can be depicted in **Figure 17** on the basis of the Mohr-Coulomb strength criterion. The linear Mohr's envelope of the weathered granite soil in this work can be expressed as

$$
\tau\_{\rm f} = \sigma \tan 41.3^{\circ} + 0.55 \,\text{MPa},\tag{5}
$$

where *τ*<sup>f</sup> is the peak shear stress and *σ* is the normal stress.

Eq. (5) shows that the cohesive strength of the samples is 55 kPa. However, because the weathered granite soil belongs to coarse grained soil, the cohesive strengths of the samples should be close to zero. It has been proved that the particles of coarse grained soil might be broken and its peak strength is nonlinear [19]. **Figure 18** shows that the ratio (*τ*f/*σ*) is not constant but decreases when *σ* increases. In addition, the Mohr envelope is bending downward. The radius of bend reduces as the confining pressure increases, which reflects that the particle breakage is controlled by the confining pressures noticeably.

According to De-Mello [20], the power function of weathered granite soil in this study can be expressed as follows:

$$
\pi\_{\rm f} = 0.361 \sigma^{0.803} \tag{6}
$$

As shown in **Figure 18**, the fitting envelope has a good correlation, which further explains that weathered granite soil has the property of nonlinear strength because of particle breakage. The nonlinear strength characteristics of compacted weathered granite soils could be affected by rock strength, intergranular friction, occlusal effect, etc. The mechanism of the nonlinear strength could be interpreted by means of the variation of failure surface angle in previous literature [21]. Under lower confining pressure, the internal friction angle was relatively large, the bending of the strength envelope was obvious, and the samples had the obvious characteristics of shear dilation and particle breakage. With the increase of confining pressure, the internal friction angle decreased and the bending degree of strength envelope decreased gradually. It was showed that under higher confining pressure, the strength characteristics of compacted weathered granite soils were mainly dominated by particle breakage.

**Figure 17.** *Mohr strength envelope of red-brown samples.*

be crushed, filled, and compacted under higher confining pressure. It was eventually manifested that the specimens transit gradually from strain softening status (e.g., when the initial confining pressure was 100 kPa) to strain hardening status (e.g., when the confining pressure was 200 kPa) with the increase of confining pressures on the samples. At any event, there exists a critical confining pressure. The samples soften in strain when the experiment confining pressures were lower than the critical confining pressures and harden in strain when the experimental confining pressures were higher than the critical confining pressures [18]. Thereafter, it could be deduced that the critical confining pressure in this study was between 100 and 200 kPa. It can be observed from **Figure 16** that the samples transit generally from shear dilation to shear shrinkage, and the volumetric strain of the transition point of the specimens increases gradually with the increase of confining pressures on the samples. In summary, as the confining pressure increases, the volumetric strain at failure changes toward volume contraction. According to

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**Figure 15.** *Stress-strain curves.*

**Figure 16.** *Curves of εv-ε1.*

**154**

phenomenon is that once the soil reaches to a certain compaction rate, it gets difficult to compact further. (2) When the confining pressures change from 100 to 300 kPa, the strain increment ratio decreases as shear stress increases. The samples reached the stage transition point (*dε*<sup>v</sup> = 0) when the strain increment ratio was equal to zero at certain level of stress. After the stage transition point, with the increase of shear force, the *dε*v/*dε*<sup>1</sup> ratio becomes negative. At this time, the volumetric strain of weathered granite soil changes toward volumetric contraction dilatancy. The aforementioned phenomena reflect the characteristics of dilatancy under lower confining pressures. The cause of the phenomena is due to the situation that the internal particles of weathered granite soil might be overturned and dislocated. (3) When the confining pressure is equal to 400 kPa, the *dε*v/*dε*<sup>1</sup> ratio is always larger than zero during shearing processes. It indicates that interparticle contact stresses in the weathered granite soil increase gradually and the particle breakage becomes obvious under larger confining pressures. At this stage, the samples were made further denser as the broken particles filled into the voids of soil

Based on the above test results, there must exist a confining pressure between 300 and 400 kPa that makes the volumetric strain increment to be equal to zero. In other words, between the aforementioned observations (2) and (3), there must exist a relatively balanced state when the dilatancy caused by the overturn and dislocation of soil particles is equal to the shrinkage caused by the breakage and

The curves of *σ*1/*σ*<sup>3</sup> versus *ε*<sup>v</sup> in **Figure 20** show the relationships between *σ*1/*σ*<sup>3</sup> and *ε*v. The relationships include two cases. One case is that the curve bends to left. The other case is that the curve bends to right. The volumetric strain of samples changes to the volume contraction initially and dilatancy later with the increase of the stress ratio of *q*/*p* under lower confining pressures, as shown in the former case when the curve bends to left. The volumetric strain of samples changes always toward the volume dilatancy with the increase of the stress ratio of *q*/*p* under larger

confining pressures in the latter case when the curve bends to right.

and the shrinkage behavior of soil samples is obvious.

compaction of soil particles.

*Weathered Granite Soils*

*DOI: http://dx.doi.org/10.5772/intechopen.86430*

**Figure 20.**

**157**

*Curves of σ1/σ<sup>3</sup> versus εv.*

**Figure 18.** *Actual strength envelope of red-brown samples.*

#### **5.4 Shearing-dilatancy characteristics**

Weathered granite soil, as coarse grained soil, has obvious shear dilation property, which was found through the CD tests in this chapter. **Figure 19** shows the results of triaxial tests under difference confining pressures with respect to the relation between the stress ratio of *q*/*p* and the strain increment ratio of *dε*v/*dε*1. The tests were performed according to the stress and strain increment parameters used in widely accepted models, such as Cam-clay model, where *q* = *σ*<sup>1</sup> *σ*<sup>3</sup> and *p* = (*σ*<sup>1</sup> + 2*σ*3)/3. **Figure 19** shows the following observations: (1) under difference confining pressures, the strain increment ratio is always larger than zero at the initial point of shear. The ratio decreases with the increase of stress ratio *q*/*p*, which indicates that the volumetric strain of weathered granite soil changes toward the volume contraction. At the same time, the contraction ratio in the initial stage of triaxial shear is larger than that in the following stage. The reason for this

**Figure 19.** *q/p: dεv/dε<sup>1</sup> of sample data with different confining pressures.*

#### *Weathered Granite Soils DOI: http://dx.doi.org/10.5772/intechopen.86430*

phenomenon is that once the soil reaches to a certain compaction rate, it gets difficult to compact further. (2) When the confining pressures change from 100 to 300 kPa, the strain increment ratio decreases as shear stress increases. The samples reached the stage transition point (*dε*<sup>v</sup> = 0) when the strain increment ratio was equal to zero at certain level of stress. After the stage transition point, with the increase of shear force, the *dε*v/*dε*<sup>1</sup> ratio becomes negative. At this time, the volumetric strain of weathered granite soil changes toward volumetric contraction dilatancy. The aforementioned phenomena reflect the characteristics of dilatancy under lower confining pressures. The cause of the phenomena is due to the situation that the internal particles of weathered granite soil might be overturned and dislocated. (3) When the confining pressure is equal to 400 kPa, the *dε*v/*dε*<sup>1</sup> ratio is always larger than zero during shearing processes. It indicates that interparticle contact stresses in the weathered granite soil increase gradually and the particle breakage becomes obvious under larger confining pressures. At this stage, the samples were made further denser as the broken particles filled into the voids of soil and the shrinkage behavior of soil samples is obvious.

Based on the above test results, there must exist a confining pressure between 300 and 400 kPa that makes the volumetric strain increment to be equal to zero. In other words, between the aforementioned observations (2) and (3), there must exist a relatively balanced state when the dilatancy caused by the overturn and dislocation of soil particles is equal to the shrinkage caused by the breakage and compaction of soil particles.

The curves of *σ*1/*σ*<sup>3</sup> versus *ε*<sup>v</sup> in **Figure 20** show the relationships between *σ*1/*σ*<sup>3</sup> and *ε*v. The relationships include two cases. One case is that the curve bends to left. The other case is that the curve bends to right. The volumetric strain of samples changes to the volume contraction initially and dilatancy later with the increase of the stress ratio of *q*/*p* under lower confining pressures, as shown in the former case when the curve bends to left. The volumetric strain of samples changes always toward the volume dilatancy with the increase of the stress ratio of *q*/*p* under larger confining pressures in the latter case when the curve bends to right.

**Figure 20.** *Curves of σ1/σ<sup>3</sup> versus εv.*

**5.4 Shearing-dilatancy characteristics**

*Actual strength envelope of red-brown samples.*

**Figure 18.**

**Figure 19.**

**156**

*q/p: dεv/dε<sup>1</sup> of sample data with different confining pressures.*

Weathered granite soil, as coarse grained soil, has obvious shear dilation property, which was found through the CD tests in this chapter. **Figure 19** shows the results of triaxial tests under difference confining pressures with respect to the relation between the stress ratio of *q*/*p* and the strain increment ratio of *dε*v/*dε*1. The tests were performed according to the stress and strain increment parameters used

in widely accepted models, such as Cam-clay model, where *q* = *σ*<sup>1</sup> *σ*<sup>3</sup> and *p* = (*σ*<sup>1</sup> + 2*σ*3)/3. **Figure 19** shows the following observations: (1) under difference confining pressures, the strain increment ratio is always larger than zero at the initial point of shear. The ratio decreases with the increase of stress ratio *q*/*p*, which indicates that the volumetric strain of weathered granite soil changes toward the volume contraction. At the same time, the contraction ratio in the initial stage of triaxial shear is larger than that in the following stage. The reason for this

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*<sup>q</sup>*<sup>c</sup> <sup>¼</sup> *Mp*�*<sup>n</sup>*

It can be seen that the product of *M*<sup>f</sup> and *M*<sup>c</sup> is constant, as shown in Eq. (11).

*<sup>M</sup>*f*M*<sup>c</sup> <sup>¼</sup> *<sup>M</sup>*<sup>2</sup>

The stress ratio at shear failure (*M*f) and the stress ratio at characteristic state point (*M*c) of granite weathered soil were calculated by a large-scale triaxial test under different confining pressures. **Figure 22** shows the shape of *M*<sup>f</sup> and *M*<sup>c</sup> in *p*-*q* plane. In this figure, *M* is the strength line determined by Mohr-Coulomb strength criterion and *M*<sup>f</sup> is the actual peak strength line. From the figure, it is found that the Mohr-Coulomb strength of compacted weathered granite soil under lower confining pressure is smaller than that of the actual strength, while the strength of compacted weathered granite soil under higher confining pressure is higher than that of the actual strength. In other words, By using Mohr-Coulomb criterion, the strength of compacted weathered granite soil under lower confining pressure may be underestimated, while that of compacted weathered granite soil under higher confining pressure may be overestimated. The phenomenon of particle breakage has been found when the triaxial test for weathered granite soil in Ube, Japan, under 10–300 kPa confining pressure has been carried out by Miura et al. [23]. It can be considered that the inapplicability of the Mohr-Coulomb strength criterion for the

By fitting the stress ratio (*M*, *M*f, and *M*c) of weathered granite soil as shown in **Figure 9**, it is found that the expression by using Eqs. (12) and (13) is more accurate

> *p*c �4*<sup>n</sup>*

> > *p*c *<sup>n</sup>*

*<sup>M</sup>*<sup>f</sup> <sup>¼</sup> *<sup>M</sup> <sup>p</sup>*

*<sup>M</sup>*<sup>c</sup> <sup>¼</sup> *<sup>M</sup> <sup>p</sup>*

weathered granite soil is caused by the particle breakage.

than by using Eqs. (7) and (8)

*Weathered Granite Soils*

*DOI: http://dx.doi.org/10.5772/intechopen.86430*

**Figure 22.**

**159**

*Shape of* M*<sup>f</sup> and* M*<sup>c</sup> on* p*-*q *plane for samples.*

<sup>c</sup> *p*1þ*n:* (10)

*,* (11)

*,* (12)

*:* (13)

**Figure 21.** *Curve of qf/p versus* �*(dεv/dε1)f.*

**Figure 21** shows the relationship between peak stress ratio and peak strain increment ratio. As shown in **Figure 21**, an approximate linear relationship can be found between *q*f/*p* and �(*dε*v/*dε*1)f. When the elastic deformation is neglected (i.e., *dεv=dε<sup>s</sup>* ¼ *dε p <sup>v</sup>=dε p <sup>s</sup>* ), *M* (*q*f/*p*) is equal to 1.83 based on the stress-dilatancy equation in a Cam-clay model [22].

#### **6. Constitutive model**

#### **6.1 Strength condition**

Particle breakage occurs in general sand under higher confining pressure and in weathered granite soil under lower confining pressure [23]. According to the generalized nonlinear strength theory proposed by Yao et al. [24], *M*<sup>f</sup> and *M*<sup>c</sup> are expressed as follows [25, 26]:

$$M\_{\rm f} = M \left(\frac{p}{p\_c}\right)^{-n},\tag{7}$$

$$M\_{\mathbf{c}} = M \left( \frac{p}{p\_{\mathbf{c}}} \right)^{n},\tag{8}$$

where *M* is the stress ratio at critical state, *M*<sup>f</sup> is the stress ratio at shear failure, *M*<sup>c</sup> is the stress ratio at characteristic state point, *p*<sup>c</sup> is the reference breaking stress, and *n* is the material parameter. Substituting *M*<sup>f</sup> = *q*f/*p* and *M*<sup>c</sup> = *q*c/*p* into Eqs. (7) and (8), respectively, gets the function of *q*<sup>f</sup> and *q*<sup>c</sup> as Eqs. (9) and (10) on the p-q plane, which are the exponential functions:

$$q\_{\rm f} = Mp\_{\rm c}^{n}p^{1-n},\tag{9}$$

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$$q\_{\mathbf{c}} = M p\_{\mathbf{c}}^{-n} p^{1+n}.\tag{10}$$

It can be seen that the product of *M*<sup>f</sup> and *M*<sup>c</sup> is constant, as shown in Eq. (11).

$$\mathbf{M}\_{\mathbf{f}} \mathbf{M}\_{\mathbf{c}} = \mathbf{M}^2,\tag{11}$$

The stress ratio at shear failure (*M*f) and the stress ratio at characteristic state point (*M*c) of granite weathered soil were calculated by a large-scale triaxial test under different confining pressures. **Figure 22** shows the shape of *M*<sup>f</sup> and *M*<sup>c</sup> in *p*-*q* plane. In this figure, *M* is the strength line determined by Mohr-Coulomb strength criterion and *M*<sup>f</sup> is the actual peak strength line. From the figure, it is found that the Mohr-Coulomb strength of compacted weathered granite soil under lower confining pressure is smaller than that of the actual strength, while the strength of compacted weathered granite soil under higher confining pressure is higher than that of the actual strength. In other words, By using Mohr-Coulomb criterion, the strength of compacted weathered granite soil under lower confining pressure may be underestimated, while that of compacted weathered granite soil under higher confining pressure may be overestimated. The phenomenon of particle breakage has been found when the triaxial test for weathered granite soil in Ube, Japan, under 10–300 kPa confining pressure has been carried out by Miura et al. [23]. It can be considered that the inapplicability of the Mohr-Coulomb strength criterion for the weathered granite soil is caused by the particle breakage.

By fitting the stress ratio (*M*, *M*f, and *M*c) of weathered granite soil as shown in **Figure 9**, it is found that the expression by using Eqs. (12) and (13) is more accurate than by using Eqs. (7) and (8)

$$M\_{\rm f} = M \left(\frac{p}{p\_c}\right)^{-4n},\tag{12}$$

$$M\_{\odot} = M \left( \frac{p}{p\_c} \right)^n. \tag{13}$$

**Figure 22.** *Shape of* M*<sup>f</sup> and* M*<sup>c</sup> on* p*-*q *plane for samples.*

**Figure 21** shows the relationship between peak stress ratio and peak strain increment ratio. As shown in **Figure 21**, an approximate linear relationship can be found between *q*f/*p* and �(*dε*v/*dε*1)f. When the elastic deformation is neglected

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Particle breakage occurs in general sand under higher confining pressure and in weathered granite soil under lower confining pressure [23]. According to the generalized nonlinear strength theory proposed by Yao et al. [24], *M*<sup>f</sup> and *M*<sup>c</sup> are

> *p*c �*<sup>n</sup>*

*p*c *<sup>n</sup>*

where *M* is the stress ratio at critical state, *M*<sup>f</sup> is the stress ratio at shear failure, *M*<sup>c</sup> is the stress ratio at characteristic state point, *p*<sup>c</sup> is the reference breaking stress, and *n* is the material parameter. Substituting *M*<sup>f</sup> = *q*f/*p* and *M*<sup>c</sup> = *q*c/*p* into Eqs. (7) and (8), respectively, gets the function of *q*<sup>f</sup> and *q*<sup>c</sup> as Eqs. (9) and (10) on the p-q

*,* (7)

*,* (8)

<sup>c</sup>*p*<sup>1</sup>�*n,* (9)

*<sup>M</sup>*<sup>f</sup> <sup>¼</sup> *<sup>M</sup> <sup>p</sup>*

*<sup>M</sup>*<sup>c</sup> <sup>¼</sup> *<sup>M</sup> <sup>p</sup>*

*<sup>q</sup>*<sup>f</sup> <sup>¼</sup> *Mpn*

*<sup>s</sup>* ), *M* (*q*f/*p*) is equal to 1.83 based on the stress-dilatancy

(i.e., *dεv=dε<sup>s</sup>* ¼ *dε*

**Figure 21.**

**158**

**6. Constitutive model**

*Curve of qf/p versus* �*(dεv/dε1)f.*

**6.1 Strength condition**

expressed as follows [25, 26]:

plane, which are the exponential functions:

*p <sup>v</sup>=dε p*

equation in a Cam-clay model [22].

According to Eq. (14),

$$M\_{\rm f}M\_{\rm c} = M^2 \left(\frac{p}{p\_{\rm c}}\right)^{-3n},\tag{14}$$

*<sup>f</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a

*DOI: http://dx.doi.org/10.5772/intechopen.86430*

function, *ε*

p

*Weathered Granite Soils*

*6.2.2 Hardening parameter*

weathered granite soil.

*6.2.3 Constitutive relation*

*∂f*

Eqs. (24)–(26):

and (27)

**161**

breakage in Eq. (22), we obtain,

*<sup>∂</sup><sup>p</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a

*∂f*

d*ε* p

<sup>v</sup> <sup>¼</sup> *m C*ð Þ <sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a

> ð Þ <sup>2</sup>*n*þ<sup>1</sup> *<sup>p</sup>*2*n*þ<sup>1</sup> *M*<sup>2</sup> c

*<sup>∂</sup><sup>q</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a

ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*2*<sup>n</sup>* c

*q*2 *p*

( )

The above expression is the yield function of weathered granite soil. In the

� � *<sup>m</sup>*

<sup>þ</sup> *<sup>p</sup>*2*n*þ<sup>1</sup>

On the basis of existing research results, Yao et al. [29] proposed the hardening

*M*<sup>4</sup> <sup>f</sup> � *<sup>η</sup>*<sup>4</sup> � � *M*<sup>4</sup> <sup>c</sup> � *<sup>η</sup>*<sup>4</sup> � � <sup>d</sup>*ε*<sup>p</sup>

*M*<sup>4</sup> <sup>c</sup> � *<sup>η</sup>*<sup>4</sup> � � *M*<sup>4</sup>

Eq. (23) is used as a hardening parameter to construct the constitutive model of

By using the theory of plastic displacement potential and the associated flow rule

function, Eq. (21), was replaced by the hardening parameter *H* considering particle

<sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup>

*q*2 *p*

According to the definition of generalized deviant stress, the plastic volumetric

The stress-dilatancy equation is expressed in the following form from Eqs. (17)

<sup>f</sup> � *<sup>η</sup>*<sup>4</sup> � � <sup>d</sup>*<sup>p</sup>* <sup>þ</sup>

<sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup> � �*<sup>m</sup>*�2*n*�<sup>1</sup> <sup>2</sup>*n*þ<sup>1</sup> <sup>2</sup>*p*<sup>2</sup>*<sup>n</sup>*

c

2*n*þ1

<sup>v</sup> is an independent variable and can be used as the hardening parameter

� *pm* 0

� *<sup>ε</sup><sup>p</sup>*

<sup>v</sup> ¼ 0*:* (21)

<sup>v</sup>*:* (22)

p

<sup>v</sup> in the yield

� �*,* (25)

(27)

<sup>f</sup> � *<sup>η</sup>*<sup>4</sup> � � <sup>d</sup>*H:* (23)

*<sup>p</sup>*<sup>2</sup>*<sup>n</sup>* � *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>* c *M*<sup>2</sup> *q*2 *p*2 � �*,* (24)

*<sup>∂</sup><sup>H</sup>* ¼ �<sup>1</sup> (26)

2*η M*<sup>2</sup>

!

*<sup>p</sup>*<sup>2</sup>*<sup>n</sup>* � *<sup>p</sup>*<sup>2</sup>*<sup>n</sup> M*<sup>2</sup> c *η*2

!

<sup>c</sup> � *<sup>η</sup>*<sup>2</sup> <sup>d</sup>*<sup>q</sup>*

*:*

c *M*<sup>2</sup> *q p*

*M*<sup>2</sup>

of the constitutive model of weathered granite soil.

parameters considering particle breakage as follows:

*H* ¼

Simplifying the differential form of Eq. (21), we obtain

d*ε*<sup>p</sup> <sup>v</sup> <sup>¼</sup> *<sup>M</sup>*<sup>4</sup> f *M*<sup>4</sup> c

and using the basic assumptions in the Cambridge model, after d*ε*

c

*<sup>m</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>*

*M*<sup>2</sup>

*∂f*

strain can be expressed as follows by means of a series of transformations of

*M*<sup>4</sup> <sup>c</sup> � *<sup>η</sup>*<sup>4</sup> � � *M*<sup>4</sup>

*M*<sup>4</sup> f *M*<sup>4</sup> c

*<sup>η</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup> h i*<sup>m</sup>*�2*n*�<sup>1</sup> <sup>2</sup>*n*þ<sup>1</sup>

*q*2 *p*

� �*<sup>m</sup>*�2*n*�<sup>1</sup> <sup>2</sup>*n*þ<sup>1</sup>

*<sup>m</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>*

*M*<sup>2</sup>

ð *M*<sup>4</sup> c *M*<sup>4</sup> f

we can see that the product of *M*<sup>f</sup> and *M*<sup>c</sup> is not a constant, but a power function of *p*.

#### **6.2 Constitutive model**

#### *6.2.1 Yield function*

Based on the results of the isotropic consolidation test for Toyoura sand, Nakai [27] has established the relationship between the volumetric strain (including elastic volumetric strain and plastic volumetric strain) and the mean principal stress of sand,

$$\epsilon\_\mathbf{v}^\mathbf{e} = \mathbf{C}\_\mathbf{e} \left[ \left( \frac{p}{p\_\mathbf{a}} \right)^m - \left( \frac{p\_0}{p\_\mathbf{a}} \right)^m \right], \tag{15}$$

$$\epsilon\_\mathbf{v}^\mathbf{p} = (\mathbf{C}\_\mathbf{t} - \mathbf{C}\_\mathbf{e}) \left[ \left( \frac{p}{p\_\mathbf{a}} \right)^m - \left( \frac{p\_\mathbf{o}}{p\_\mathbf{a}} \right)^m \right], \tag{16}$$

where *p*<sup>0</sup> is the initial mean principal stress, *p*<sup>a</sup> (=101.3 kPa) is the atmospheric pressure, *m* is a coefficient for sand, *C*<sup>t</sup> is the compression index, and *C*<sup>c</sup> is the swelling index. Based on the Eqs. (15) and (16), the yield function of weathered granite soil is constructed in this chapter.

When considering the particle breakage of soil, the stress-dilatancy equation in the modified Cambridge model was changed by Yao et al. [28] to the following form:

$$\frac{\mathrm{d}\varepsilon\_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}\varepsilon\_{\mathrm{s}}^{\mathrm{p}}} = \frac{M\_{\mathrm{c}}^{2} - \eta^{2}}{2\eta},\tag{17}$$

where *ε* p <sup>s</sup> is the plastic shear strain and *η* is *q*/*p*. The orthogonality condition is

$$-\frac{\mathrm{d}\varepsilon\_{\mathrm{V}}^{\mathrm{P}}}{\mathrm{d}\varepsilon\_{\mathrm{s}}^{\mathrm{P}}} = \frac{\mathrm{d}q}{\mathrm{d}p},\tag{18}$$

The expression of the plastic potential function, which is obtained through the differential equation, composed of Eqs. (17) and (18), is expressed as

$$f = (2n+1)\frac{p\_c^{2n}q^2}{M^2} + p^{2n+1} - p\_x^{2n+1} = 0.\tag{19}$$

According to Nakai's research [27], *p*<sup>x</sup> is expressed as

$$p\_{\mathbf{x}} = \left(\frac{p\_{\mathbf{a}}^{m}}{C\_{\mathbf{t}} - C\_{\mathbf{e}}} \epsilon\_{\mathbf{v}}^{\mathbf{p}} + p\_{\mathbf{0}}^{m}\right)^{\frac{1}{m}}.\tag{20}$$

Substituting the Eq. (20) into the Eq. (19), we get

*Weathered Granite Soils DOI: http://dx.doi.org/10.5772/intechopen.86430*

$$f = \frac{C\_{\mathbf{t}} - C\_{\mathbf{e}}}{p\_{\mathbf{s}}^m} \left\{ \left[ \frac{(2n+1)p\_{\mathbf{c}}^{2n}}{M^2} \frac{q^2}{p} + p^{2n+1} \right]^{\frac{m}{2n+1}} - p\_0^m \right\} - \epsilon\_{\mathbf{v}}^p = \mathbf{0}. \tag{21}$$

The above expression is the yield function of weathered granite soil. In the function, *ε* p <sup>v</sup> is an independent variable and can be used as the hardening parameter of the constitutive model of weathered granite soil.

#### *6.2.2 Hardening parameter*

According to Eq. (14),

function of *p*.

**6.2 Constitutive model**

*6.2.1 Yield function*

of sand,

form:

**160**

where *ε* p *<sup>M</sup>*f*M*<sup>c</sup> <sup>¼</sup> *<sup>M</sup>*<sup>2</sup> *<sup>p</sup>*

*Geotechnical Engineering - Advances in Soil Mechanics and Foundation Engineering*

we can see that the product of *M*<sup>f</sup> and *M*<sup>c</sup> is not a constant, but a power

Based on the results of the isotropic consolidation test for Toyoura sand, Nakai [27] has established the relationship between the volumetric strain (including elastic volumetric strain and plastic volumetric strain) and the mean principal stress

> *p p*a *<sup>m</sup>*

� *<sup>p</sup>*<sup>0</sup> *p*a

*<sup>m</sup>*

� *<sup>p</sup>*<sup>0</sup> *p*a

*<sup>m</sup>*

*p p*a *<sup>m</sup>*

where *p*<sup>0</sup> is the initial mean principal stress, *p*<sup>a</sup> (=101.3 kPa) is the atmospheric pressure, *m* is a coefficient for sand, *C*<sup>t</sup> is the compression index, and *C*<sup>c</sup> is the swelling index. Based on the Eqs. (15) and (16), the yield function of weathered

When considering the particle breakage of soil, the stress-dilatancy equation in the modified Cambridge model was changed by Yao et al. [28] to the following

<sup>c</sup> � *<sup>η</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>M</sup>*<sup>2</sup>

*ε*e <sup>v</sup> ¼ *C*<sup>e</sup>

<sup>v</sup> ¼ ð Þ *C*<sup>t</sup> � *C*<sup>e</sup>

d*ε* p v d*ε* p s

> � d*ε* p v d*ε* p s <sup>¼</sup> <sup>d</sup>*<sup>q</sup>*

differential equation, composed of Eqs. (17) and (18), is expressed as

*px* <sup>¼</sup> *<sup>p</sup><sup>m</sup>*

a *C*<sup>t</sup> � *C*<sup>e</sup> *ε*p <sup>v</sup> <sup>þ</sup> *pm* 0

<sup>1</sup>

c *M*<sup>2</sup> *q*2 *p*

The expression of the plastic potential function, which is obtained through the

<sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup> � *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup>

*m*

<sup>s</sup> is the plastic shear strain and *η* is *q*/*p*.

*<sup>f</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>*

According to Nakai's research [27], *p*<sup>x</sup> is expressed as

Substituting the Eq. (20) into the Eq. (19), we get

*ε*p

granite soil is constructed in this chapter.

The orthogonality condition is

*p*c �3*<sup>n</sup>*

*,* (14)

*,* (15)

<sup>2</sup>*<sup>η</sup> ,* (17)

<sup>d</sup>*<sup>p</sup> ,* (18)

*<sup>x</sup>* ¼ 0*:* (19)

*:* (20)

*,* (16)

On the basis of existing research results, Yao et al. [29] proposed the hardening parameters considering particle breakage as follows:

$$H = \int \frac{M\_\mathrm{c}^4}{M\_\mathrm{f}^4} \frac{\left(M\_\mathrm{f}^4 - \eta^4\right)}{\left(M\_\mathrm{c}^4 - \eta^4\right)} \mathrm{d}\varepsilon\_\mathrm{v}^\mathrm{P}.\tag{22}$$

Simplifying the differential form of Eq. (21), we obtain

$$\mathrm{d}\epsilon\_{\mathrm{v}}^{\mathrm{p}} = \frac{\mathcal{M}\_{\mathrm{f}}^{4}}{\mathcal{M}\_{\mathrm{c}}^{4}} \frac{\left(\mathcal{M}\_{\mathrm{c}}^{4} - \eta^{4}\right)}{\left(\mathcal{M}\_{\mathrm{f}}^{4} - \eta^{4}\right)} \mathrm{d}H.\tag{23}$$

Eq. (23) is used as a hardening parameter to construct the constitutive model of weathered granite soil.

#### *6.2.3 Constitutive relation*

By using the theory of plastic displacement potential and the associated flow rule and using the basic assumptions in the Cambridge model, after d*ε* p <sup>v</sup> in the yield function, Eq. (21), was replaced by the hardening parameter *H* considering particle breakage in Eq. (22), we obtain,

$$\frac{\partial \mathcal{f}}{\partial p} = \frac{\mathbf{C\_t} - \mathbf{C\_e}}{p\_\mathbf{a}^m} m \left[ \frac{(2n+1)p\_c^{2n}}{M^2} \frac{q^2}{p} + p^{2n+1} \right]^{\frac{m-2n-1}{2n+1}} \left( p^{2n} - \frac{p\_c^{2n}}{M^2} \frac{q^2}{p^2} \right), \tag{24}$$

$$\frac{\partial \mathcal{f}}{\partial q} = \frac{\mathcal{C}\_{\text{t}} - \mathcal{C}\_{\text{e}}}{p\_{\text{a}}^{m}} m \left[ \frac{(2n+1)p\_{\text{c}}^{2n}}{M^{2}} \frac{q^{2}}{p} + p^{2n+1} \right]^{\frac{m-2n-1}{2n+1}} \left( \frac{2p\_{\text{c}}^{2n}}{M^{2}} \frac{q}{p} \right),\tag{25}$$

$$\frac{\partial f}{\partial H} = -\mathbf{1} \tag{26}$$

According to the definition of generalized deviant stress, the plastic volumetric strain can be expressed as follows by means of a series of transformations of Eqs. (24)–(26):

$$\begin{split} \mathrm{d}e\_{\mathrm{v}}^{\mathrm{p}} &= \frac{m(\mathcal{C}\_{\mathrm{t}} - \mathcal{C}\_{\mathrm{e}})}{p\_{\mathrm{a}}^{m}} \frac{\mathcal{M}\_{\mathrm{f}}^{4}}{\mathcal{M}\_{\mathrm{c}}^{4}} \frac{\left(\mathcal{M}\_{\mathrm{c}}^{4} - \eta^{4}\right)}{\left(\mathcal{M}\_{\mathrm{f}}^{4} - \eta^{4}\right)} \left(\mathrm{d}p + \frac{2\eta}{\mathcal{M}\_{\mathrm{c}}^{2} - \eta^{2}} \mathrm{d}q\right) \\ &\qquad \left[\frac{(2n+1)p^{2n+1}}{\mathcal{M}\_{\mathrm{c}}^{4}} \eta^{2} + p^{2n+1}\right]^{\frac{n-2n-1}{2n+1}} \left(p^{2n} - \frac{p^{2n}}{\mathcal{M}\_{\mathrm{c}}^{2}} \eta^{2}\right). \end{split} \tag{27}$$

The stress-dilatancy equation is expressed in the following form from Eqs. (17) and (27)

*Geotechnical Engineering - Advances in Soil Mechanics and Foundation Engineering*

$$\begin{split} \mathbf{d}e\_{\mathbf{s}}^{\mathbf{p}} &= \frac{m(\mathbf{C}\_{\mathbf{t}} - \mathbf{C}\_{\mathbf{e}})}{p\_{\mathbf{s}}^{m}} \frac{M\_{\mathbf{f}}^{4}}{M\_{\mathbf{c}}^{4}} \frac{(\mathcal{M}\_{\mathbf{c}}^{2} + \eta^{2}) 2\eta}{(\mathcal{M}\_{\mathbf{f}}^{4} - \eta^{4})} \left(\mathbf{d}p + \frac{2\eta}{M\_{\mathbf{c}}^{2} - \eta^{2}} \mathbf{d}q\right) . \\ &\left[\frac{(2n+1)p^{2n+1}}{M\_{\mathbf{c}}^{2}}\eta^{2} + p^{2n+1}\right]^{\frac{m-2n-1}{2n+1}} \left(p^{2n} - \frac{p^{2n}}{M\_{\mathbf{c}}^{2}}\eta^{2}\right) \end{split} \tag{28}$$

Equations (26) and (27) are the plastic strain expression of weathered granite soil considering particle breakage.

In three-dimensional axisymmetry, the following expressions can be obtained from the generalized Hook's law:

$$\mathrm{d}\boldsymbol{\varepsilon}\_{\mathrm{v}}^{\mathrm{e}} = \frac{\mathfrak{Z}(1-2\boldsymbol{\nu})}{E} \mathrm{d}\boldsymbol{p} \tag{29}$$

**7. Conclusions**

*Weathered Granite Soils*

*DOI: http://dx.doi.org/10.5772/intechopen.86430*

granite [11].

per layer, and stress level.

as below:

In this chapter, routine physical and mechanics tests and large-scale triaxial tests

1. With an increase in burying depth of weathered granite, the geological year's parameter (*m*) decreases by power function, but the geometric progression constants (*r*) increase by power function. The exponent of power function in this chapter can be used to evaluate the weathering process of weathered

2. The particle-breaking characteristics of weathered granite soils are obviously influenced by many factors such as particle gradation, mineral content, blows

3. The pure weathered granite soil shows similar compaction characteristics of sands. The compression and bearing characteristics of weathered granite soil vary significantly when the clay content ratio changes. The weathered granite

4.The experimental results show that the increase in peak deviatoric stress *q*<sup>f</sup> due to an increase in mean stress *p* is observed as nonlinear under lower confining pressure because of the existence of particle breakage. It is found that the product of the state stress ratio and peak stress ratio is not a constant but a power function of an average main stress. The research results are helpful to understand the law of the long-term degradation of subgrade performance due

Given that wet conditions may influence the evolution of embankments after construction, future extensive research should be focused on dynamic measurement methods of particle breakage and long-term behavior degradation analysis of subgrade as a result of further weathering and particle breakage [7, 11, 30, 31]. It is very difficult to entirely understand the road performance of weathered granite soil due to its special mechanical properties. Further research needs to be performed for assessing the unique mechanical behaviors of compacted weathered granite soil considering particle breakage and the mechanical properties of different types of weathered granites soils (e.g., coarse-grained weathered granite soil and finegrained weathered granite soil). Given that measuring particle breakage and its variations during triaxial compression tests are still a challenging task [32], future extensive research should be focused on the dynamic measurement methods, the mathematical description of particle breakage, and the long-term behavior degra-

The study in this chapter was sponsored by the China Postdoctoral Science Foundation (2016M591044), the National Basic Research Program of China (973 Program, 2014CB047006), the Research Plan of Shanxi Province Department of

soil in this chapter can be used as the filling material.

to particle breakage of weathered granite soil.

dation owning to particle breakage [33].

**Acknowledgements**

**163**

were conducted to investigate the compaction, bearing, strength and shearingdilatancy characteristics, and constitutive model of compacted weathered granite soil. The main conclusions obtained from the study in this chapter are summarized

$$\mathrm{d}\varepsilon\_{\mathrm{s}}^{\mathrm{e}} = \frac{2(1+\nu)}{3E} \mathrm{d}p \tag{30}$$

where E is the elastic modulus and v is the Poisson's ratio.

After taking the derivative of *p* in Eq. (15), the elastic volumetric strain can be expressed as

$$\mathbf{d} \epsilon\_{\mathbf{v}}^{\mathbf{e}} = \frac{m C\_{\mathbf{e}} p^{m-1}}{p\_{\mathbf{a}}^{m}} \mathbf{d}p \tag{31}$$

Combining Eqs. (29)–(31), the elastic shear strain is obtained in the following form,

$$\mathbf{d}\epsilon\_s^\mathbf{e} = \frac{2(\mathbf{1} + \boldsymbol{\nu})m\mathbf{C}\_\mathbf{e} p^{m-1}}{\Re(\mathbf{1} - \boldsymbol{2}\boldsymbol{\nu})p\_\mathbf{a}^m} \mathbf{d}p \tag{32}$$

According to elastoplastic mechanics, the total volumetric strain and the total shear strain are, respectively, expressed as

<sup>d</sup>*ε*<sup>v</sup> <sup>¼</sup> <sup>d</sup>*ε*<sup>e</sup> <sup>v</sup> þ d*ε* p <sup>v</sup> <sup>¼</sup> *m C*ð Þ <sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a *M*<sup>4</sup> f *M*<sup>4</sup> c *M*<sup>4</sup> <sup>c</sup> � *<sup>η</sup>*<sup>4</sup> � � *M*<sup>4</sup> <sup>f</sup> � *<sup>η</sup>*<sup>4</sup> � � <sup>d</sup>*<sup>p</sup>* <sup>þ</sup> 2*η M*<sup>2</sup> <sup>c</sup> � *<sup>η</sup>*<sup>2</sup> <sup>d</sup>*<sup>q</sup>* !� ð Þ <sup>2</sup>*n*þ<sup>1</sup> *<sup>p</sup>*2*n*þ<sup>1</sup> *M*<sup>2</sup> c *<sup>η</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup> h i*<sup>m</sup>*�2*n*�<sup>1</sup> <sup>2</sup>*n*þ<sup>1</sup> *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>* � *<sup>p</sup>*<sup>2</sup>*<sup>n</sup> M*<sup>2</sup> c *η*2 ! þ *mC*e*pm*�<sup>1</sup> *pm* a d*p* (33) <sup>d</sup>*ε*<sup>s</sup> <sup>¼</sup> <sup>d</sup>*ε*<sup>e</sup> <sup>s</sup> þ d*ε* p <sup>s</sup> <sup>¼</sup> *m C*ð Þ <sup>t</sup> � *<sup>C</sup>*<sup>e</sup> *pm* a *M*<sup>4</sup> f *M*<sup>4</sup> c *M*<sup>2</sup> <sup>c</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � �2*<sup>η</sup> M*<sup>4</sup> <sup>f</sup> � *<sup>η</sup>*<sup>4</sup> � � <sup>d</sup>*<sup>p</sup>* <sup>þ</sup> 2*η M*<sup>2</sup> <sup>c</sup> � *<sup>η</sup>*<sup>2</sup> <sup>d</sup>*<sup>q</sup>* !� ð Þ <sup>2</sup>*n*þ<sup>1</sup> *<sup>p</sup>*2*n*þ<sup>1</sup> *M*<sup>2</sup> c *<sup>η</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup>*n*þ<sup>1</sup> h i*<sup>m</sup>*�2*n*�<sup>1</sup> <sup>2</sup>*n*þ<sup>1</sup> *<sup>p</sup>*<sup>2</sup>*<sup>n</sup>* � *<sup>p</sup>*<sup>2</sup>*<sup>n</sup> M*<sup>2</sup> c *η*2 ! þ 2 1ð Þ <sup>þ</sup> *<sup>v</sup> mC*e*p<sup>m</sup>*�<sup>1</sup> 9 1ð Þ � 2*v pm* a d*q:* (34)

The Eqs. (33) and (34) are the constitutive relation of the weathered granite soils considering particle brakeage.

The seven parameters in the constitutive model of weathered granite soil are *C*e, *C*t, *m*, *M*, *p*c, *n*, and *ν.* Except for Poisson's ratio *ν*, the other six parameters can be obtained by conventional triaxial test. The value of *ν* is assumed to be 0.3. *C*e, *C*t, and *m* can be obtained by isotropic compression and unloading test, and *M*, *p*c, and *n* can be obtained by consolidated drained triaxial test.
