**Table 3.**

*Specimen volumetric water contents.*

first 15 groups of specimens were used to investigate the effects of filler plastic limit on the microheave filler frost heave property, and the next 15 groups of specimens

microheave filler frost heave property. The required amounts of the specimens were calculated based on a compactness of 0.95. The specimens were then compacted in

were used to investigate the effects of filler optimal water content on the

**Specimen Initial water content ω(%)**

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

1 12.3 2 14.5 3 15.1 4 15.5 5 15.9 6 16.0 7 16.5 8 18.0 9 18.5 10 18.8 11 19.0 12 19.6 13 20.0 14 20.5 15 21.3 16 13.3 17 14.3 18 14.4 19 14.7 20 15.5 21 15.6 22 15.8 23 16.2 24 16.3 25 17.5 26 17.8 27 18.0 28 19.1 29 19.4 30 21.7

2. Effects of volumetric water content on microheave filler frost heave

Proper amounts of dry filler were mixed with various amounts of water to produce 15 groups of *ω<sup>V</sup>* specimens with various levels of initial water content. The initial water content values of the 15 groups of specimens are listed in **Table 3**. The

the specimen box for frost heave testing.

**Table 2.**

**98**

*Specimen initial water contents.*

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


**4. Analysis of test results**

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

(excluding frost heave) in mm.

filler frost heave.

filler frost heave rate increased significantly.

*Relation between the filler plastic limit and filler frost heave.*

heave rate *η*.

rate.

**Figure 3.**

**101**

filler frost heave rate *η*.

The microheave filler frost heave rate η was calculated via the following formula:

where *Δh* is the overall specimen frost heave in mm, and *Hf* is the frozen depth

1. Relation between filler plastic limit, optimal water content and microheave

**Figures 3** and **4** show that for filler water contents of *ω* ≤ *ω<sup>p</sup>* + 2 or *ω* ≤ *ω<sup>o</sup>* + 4.6, the frost heave rate *η* < 1%. However, as the water content increased further, the

1. Relation between microheave filler volumetric water content and frost heave

**Figure 5** shows the relation between the filler volumetric water content *ω<sup>v</sup>* and

**Figure 5** shows that when *ω<sup>v</sup>* ≤ 13%, the filler frost heave rate was insensitive to increases in the volumetric water content, and when *ω<sup>v</sup>* > 13%, filler frost heave

increased significantly with increasing volumetric water content.

**Figure 3** shows the relation between *ω*-*ω<sup>p</sup>* (difference between the filler initial water content *ω* and filler plastic limit water content *ωp*) and filler frost heave rate *η*. **Figure 4** shows the relation between *ω*-*ω<sup>0</sup>* (difference between the filler initial water content *ω* and filler optimal water content *ω0*) and filler frost

� 100% (2)

*<sup>η</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>h</sup> Hf*

**4.1 Effects of water content on microheave filler frost heave**

*Frost Heave Deformation Analysis Model for Microheave Filler*

#### **Table 4.**

*Specimen filler content and filler filling rate.*

#### **Figure 2.** *Loading apparatus used for the frost heave test.*

5. Effects of overlying load on microheave filler frost heave.

A specimen with a 10% filler content was prepared for the frost heave test under test conditions in which the initial water content was 15% and the overlying loads were 5, 10, 20, 30, 40, 55, 65 and 80 kPa. **Figure 2** shows the loading apparatus used in the frost heave test.

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

## **4. Analysis of test results**

The microheave filler frost heave rate η was calculated via the following formula:

$$
\eta = \frac{\Delta h}{H\_f} \times 100\% \tag{2}
$$

where *Δh* is the overall specimen frost heave in mm, and *Hf* is the frozen depth (excluding frost heave) in mm.

#### **4.1 Effects of water content on microheave filler frost heave**

1. Relation between filler plastic limit, optimal water content and microheave filler frost heave.

**Figure 3** shows the relation between *ω*-*ω<sup>p</sup>* (difference between the filler initial water content *ω* and filler plastic limit water content *ωp*) and filler frost heave rate *η*. **Figure 4** shows the relation between *ω*-*ω<sup>0</sup>* (difference between the filler initial water content *ω* and filler optimal water content *ω0*) and filler frost heave rate *η*.

**Figures 3** and **4** show that for filler water contents of *ω* ≤ *ω<sup>p</sup>* + 2 or *ω* ≤ *ω<sup>o</sup>* + 4.6, the frost heave rate *η* < 1%. However, as the water content increased further, the filler frost heave rate increased significantly.

1. Relation between microheave filler volumetric water content and frost heave rate.

**Figure 5** shows the relation between the filler volumetric water content *ω<sup>v</sup>* and filler frost heave rate *η*.

**Figure 5** shows that when *ω<sup>v</sup>* ≤ 13%, the filler frost heave rate was insensitive to increases in the volumetric water content, and when *ω<sup>v</sup>* > 13%, filler frost heave increased significantly with increasing volumetric water content.

**Figure 3.** *Relation between the filler plastic limit and filler frost heave.*

5. Effects of overlying load on microheave filler frost heave.

in the frost heave test.

*Loading apparatus used for the frost heave test.*

**Figure 2.**

**100**

**Table 4.**

*Specimen filler content and filler filling rate.*

A specimen with a 10% filler content was prepared for the frost heave test under test conditions in which the initial water content was 15% and the overlying loads were 5, 10, 20, 30, 40, 55, 65 and 80 kPa. **Figure 2** shows the loading apparatus used

**Specimen Filler content (%) Filler filling rate (%)**

 1.0 0.056 2.0 0.118 2.5 0.198 2.8 0.272 3.0 0.316 3.2 0.334 5.0 0.363 5.5 0.381 6.0 0.445 9.0 0.465 11.0 0.529 16.0 0.532 18.0 0.606 40.0 0.859 45.0 0.928

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

**Figure 4.** *Relation between the filler optimal water content and filler frost heave.*

**Figure 5.** *Relation between the filler volumetric water content and filler frost heave rate.*

#### **4.2 Effects of filler on microheave filler frost heave**

1. Relation between filler content, filling rate and microheave filler frost heave.

2. Relation between filler frost heave and microheave filler frost heave.

microheave filler frost heave for each group is shown in **Figure 8**. **Figure 8** shows that when filler frost heave was below 25 cm3

25 cm<sup>3</sup>

**103**

**Figure 7.**

**Figure 6.**

*Microheave filler frost heave rate for different filler contents.*

*Frost Heave Deformation Analysis Model for Microheave Filler*

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

*Microheave filler frost heave rate for different filler filling rate.*

nificantly with filler frost heave.

Based on the test results, the relation between specimen filler frost heave and

filler exhibited no frost heave. Under this condition, filler frost heave filled pores, and there was no macroscopic filler frost heave. When filler frost heave exceeded

**4.3 Relation between overlying load and microheave filler frost heave**

, macroscopic frost heave in the microheave filler began and increased sig-

Overlying loads affect microheave filler frost heave properties in two ways: the freezing point drops with increasing external load, and overlying loads result in

, the microheave

**Figure 6** shows the microheave filler frost heave rates for different filler contents.

**Figure 6** shows that filler frost heave increased gradually with filler content. When the filler content was under 3%, the frost heave rate was approximately 0.2%, when the filler content was under 15%, the frost heave rate was under 1.0%, and when the filler content exceeded 15%, the filler frost heave sensitivity increased significantly with filler content.

**Figure 7** shows the microheave filler frost heave rates for different filler filling rates.

**Figure 7** shows that for filler filling rates below 0.18, the filler frost heave rates were under 0.2%, and when the filler filling rates were under 0.25, the filler frost heave rates were under 0.5%; filler frost heave was insensitive to increases in filler content. For filling rates exceeding 0.25, the filler frost heave rates increased significantly with the filling rate, and frost heave became sensitive. For the filling rates were under 0.37, the frost heave rates were under 1.0%.

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

**Figure 6.** *Microheave filler frost heave rate for different filler contents.*

**Figure 7.** *Microheave filler frost heave rate for different filler filling rate.*

2. Relation between filler frost heave and microheave filler frost heave.

Based on the test results, the relation between specimen filler frost heave and microheave filler frost heave for each group is shown in **Figure 8**.

**Figure 8** shows that when filler frost heave was below 25 cm3 , the microheave filler exhibited no frost heave. Under this condition, filler frost heave filled pores, and there was no macroscopic filler frost heave. When filler frost heave exceeded 25 cm<sup>3</sup> , macroscopic frost heave in the microheave filler began and increased significantly with filler frost heave.

#### **4.3 Relation between overlying load and microheave filler frost heave**

Overlying loads affect microheave filler frost heave properties in two ways: the freezing point drops with increasing external load, and overlying loads result in

**4.2 Effects of filler on microheave filler frost heave**

*Relation between the filler volumetric water content and filler frost heave rate.*

*Relation between the filler optimal water content and filler frost heave.*

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

were under 0.37, the frost heave rates were under 1.0%.

contents.

**Figure 5.**

**Figure 4.**

rates.

**102**

significantly with filler content.

1. Relation between filler content, filling rate and microheave filler frost heave.

**Figure 6** shows that filler frost heave increased gradually with filler content. When the filler content was under 3%, the frost heave rate was approximately 0.2%, when the filler content was under 15%, the frost heave rate was under 1.0%, and when the filler content exceeded 15%, the filler frost heave sensitivity increased

**Figure 7** shows the microheave filler frost heave rates for different filler filling

**Figure 7** shows that for filler filling rates below 0.18, the filler frost heave rates were under 0.2%, and when the filler filling rates were under 0.25, the filler frost heave rates were under 0.5%; filler frost heave was insensitive to increases in filler content. For filling rates exceeding 0.25, the filler frost heave rates increased significantly with the filling rate, and frost heave became sensitive. For the filling rates

**Figure 6** shows the microheave filler frost heave rates for different filler

and water content volumes. When fine grains develop frost heave, some water

*Frost Heave Deformation Analysis Model for Microheave Filler*

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

In this section, a simulation embodying the direct expansion of filler grains under natural conditions is examined to study filler grain expansion due to water adsorption. To investigate the complete filling of microheave filler, the mixed material specimen model was simplified. The model was assumed to consist of coarse skeletons, filler grains and pores, as shown in **Figures 10** and **11**.

Filler and residual pores constituted the mixed material skeleton pores. Among them, filler fills the skeleton pores, expands at low temperature, and, under the side constraint condition, fills the residual pores first. When the residual pores are filled, the expansion is represented as a mixed material specimen macroscopic expansion.

If *α* represents the filler expansion rate, *θ* represents the filler content, *ρmixed* represents the mixed material compactness, and *Gskeleton* represents the skeleton

*ρfiller*

*mfiller* <sup>¼</sup> *<sup>θ</sup>*

Δ*Vmixed material* ¼ Δ*Vfiller* � Δ*Vskeleton pore* (3)

Δ*Vskeleton pore* ¼ Δ*Vresidual pore* (4)

ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> mskeleton*

ð Þ <sup>1</sup> � *<sup>θ</sup> mskeleton* (6)

*<sup>ρ</sup>skeleton* � *Vmixed material*

(5)

content escapes.

grain ratio, then

Because

**Figure 10.** *Mixed material.*

**105**

From formula (4) and Δ*Vfiller* = *Vfillerα*,

<sup>Δ</sup>*Vmixed material* <sup>¼</sup> <sup>Δ</sup>*Vfiller* � <sup>Δ</sup>*Vresidual pore* <sup>¼</sup> *mfiller*

based on the macroscopic volume expansion rate,

**Figure 8.** *Filler frost heave versus microheave filler frost heave.*

**Figure 9.** *Relation between frost heave rate and overlying load.*

redistributions of filler water content [20]. **Figure 9** shows the microheave filler frost heave rates under different overlying loads.

**Figure 9** shows that overlying load and the microheave filler frost heave rate were exponential related. The microheave filler frost heave rate decreased gradually with increasing overlying load.

The indoor test described above provided data for a qualitative analysis of microheave filler frost heave rules. To further investigate frost heave deformation, theoretical analysis and microheave filler frost heave model creation are required.

## **5. Microheave filler frost heave model**

#### **5.1 Microheave filler complete filling analysis**

When a microheave filler specimen develops frost heave, the coarse grains do not develop frost heave and act as rigid bodies. By comparison, fine grains develop frost heave at low temperatures. Fine grain component volumes consist of soil grain *Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

and water content volumes. When fine grains develop frost heave, some water content escapes.

In this section, a simulation embodying the direct expansion of filler grains under natural conditions is examined to study filler grain expansion due to water adsorption. To investigate the complete filling of microheave filler, the mixed material specimen model was simplified. The model was assumed to consist of coarse skeletons, filler grains and pores, as shown in **Figures 10** and **11**.

Filler and residual pores constituted the mixed material skeleton pores. Among them, filler fills the skeleton pores, expands at low temperature, and, under the side constraint condition, fills the residual pores first. When the residual pores are filled, the expansion is represented as a mixed material specimen macroscopic expansion.

$$
\Delta V\_{\text{mixed material}} = \Delta V\_{\text{filter}} - \Delta V\_{\text{selection pore}} \tag{3}
$$

If *α* represents the filler expansion rate, *θ* represents the filler content, *ρmixed* represents the mixed material compactness, and *Gskeleton* represents the skeleton grain ratio, then

$$
\Delta V\_{skeleton\ pore} = \Delta V\_{residual\ pore} \tag{4}
$$

From formula (4) and Δ*Vfiller* = *Vfillerα*,

$$
\Delta V\_{\text{mixed material}} = \Delta V\_{\text{filter}} - \Delta V\_{\text{residual pore}} = \frac{m\_{\text{filter}}}{\rho\_{\text{filter}}} (1 + a) + \left(\frac{m\_{\text{selection}}}{\rho\_{\text{selection}}}\right) - V\_{\text{mixed material}}\tag{5}
$$

Because

$$m\_{filter} = \frac{\theta}{(1-\theta)} m\_{selection} \tag{6}$$

based on the macroscopic volume expansion rate,

**Figure 10.** *Mixed material.*

redistributions of filler water content [20]. **Figure 9** shows the microheave filler

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

The indoor test described above provided data for a qualitative analysis of microheave filler frost heave rules. To further investigate frost heave deformation, theoretical analysis and microheave filler frost heave model creation are required.

When a microheave filler specimen develops frost heave, the coarse grains do not develop frost heave and act as rigid bodies. By comparison, fine grains develop frost heave at low temperatures. Fine grain component volumes consist of soil grain

**Figure 9** shows that overlying load and the microheave filler frost heave rate were exponential related. The microheave filler frost heave rate decreased gradually

frost heave rates under different overlying loads.

*Relation between frost heave rate and overlying load.*

*Filler frost heave versus microheave filler frost heave.*

**5. Microheave filler frost heave model**

**5.1 Microheave filler complete filling analysis**

with increasing overlying load.

**Figure 8.**

**Figure 9.**

**104**

**Figure 11.** *Composition of the mixed material.*

$$\eta\_{\text{mixed\ material}} = \frac{\frac{\Delta V\_{\text{mixed\ material}}}{\pi R^2}}{\frac{V\_{\text{mixed\ material}}}{\pi R^2}} = \frac{\Delta V\_{\text{mixed\ material}}}{V\_{\text{mixed\ material}}} \tag{7}$$

$$
\eta\_{\text{filter}} = a \tag{8}
$$

*mskeleton* ¼ *ρ*ð Þ 1 � *θ Vmixed material* (11)

þ

" #

1 *Gskeletonρ<sup>w</sup>*

<sup>¼</sup> *mskeleton* <sup>þ</sup> *mmixed material* ð Þ 1 þ *e* ð Þ *Vskeleton* þ *Vmixed material*

*=Vmixed material* � 1

(12)

� 1 (13)

(14)

(15)

(17)

(18)

� 1 (16)

1 *ρskeleton*

1 *ρskeleton*

þ

þ

*θ* ð Þ <sup>1</sup>�*<sup>θ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> ρfiller*

Based on the definition of a mixed material skeleton pore, *e* = *Vresidual pore*/*VS*,

*<sup>ρ</sup>filler* <sup>¼</sup> *<sup>θ</sup>* 1 ð Þ 1þ*e ρmixed*

� ð Þ <sup>1</sup>�*<sup>θ</sup>* ð Þ *Gskeletonρ<sup>w</sup>*

*ρmixed*

*dsρw*ð Þ <sup>1</sup>þ0*:*001*<sup>w</sup>* ‐ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>=</sup>*ð Þ *Gskeletonρ<sup>w</sup>*

þ

*ρmixed*ð Þ 1 � *θ α Gskeletonρ<sup>w</sup>*

1 *Gskeletonρ<sup>w</sup>*

� 1

*<sup>ρ</sup>* � <sup>1</sup> <sup>¼</sup> *dsρw*ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*001*<sup>w</sup>*

<sup>¼</sup> *<sup>θ</sup>* 1

> *θ* ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> ρfiller*

� �

Combining formulas (9) and (11),

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

¼ *ρmixed*ð Þ 1 � *θ*

*θ* ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> ρfiller*

*θ* ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> ρfiller*

*Frost Heave Deformation Analysis Model for Microheave Filler*

Because the skeleton grain ratio is *Gskeleton* = *ρskeleton*/*ρw*,

*ηmixed material* ¼ *ρmixed*ð Þ 1 � *θ*

*Vskeleton* þ *Vmixed material* þ *Vresidual pore*

*<sup>ρ</sup>mixed* <sup>¼</sup> *mskeleton* <sup>þ</sup> *mmixed material*

and employing the skeleton grain ratio,

Based on the definition of the porosity ratio,

Based on formulas (15) and (16),

Based on formulas (15) and (17),

*ηmixed material* ¼ *ρmixed*ð Þ 1 � *θ*

Formula (18) is simplified to

**107**

*<sup>ρ</sup>filler* <sup>¼</sup> *<sup>θ</sup>* 1 ð Þ 1þ*e ρmixed*

*<sup>e</sup>* <sup>¼</sup> *dsρw*ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*001*<sup>w</sup>*

� ð Þ <sup>1</sup>�*<sup>θ</sup>* ð Þ *Gskeletonρ<sup>w</sup>*

<sup>¼</sup> *<sup>ρ</sup>mixed*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> dsρw*ð Þ 1 þ 0*:*001*w* ‐

*ηmixed material* ¼ *mmixed material*

from formula (12),

From formulas (7) and (8),

$$
\eta\_{\text{mixed material}} = m\_{\text{mixed material}} \left[ \frac{\frac{\theta}{(1-\theta)}(\mathbf{1}+a)}{\rho\_{\text{filter}}} + \frac{\mathbf{1}}{\rho\_{\text{selection}}} \right] / V\_{\text{mixed material}} - \mathbf{1} \tag{9}
$$

Because

$$\rho\_{\text{mixed}} = \frac{m\_{\text{skeleton}} + m\_{\text{mixed material}}}{V\_{\text{mixed material}}} \tag{10}$$

Based on formulas (6) and (10),

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

$$m\_{skeleton} = \rho(\mathbf{1} - \theta) V\_{mixed\ material} \tag{11}$$

Combining formulas (9) and (11),

$$\begin{aligned} \eta\_{\text{mixed\ material}} &= m\_{\text{mixed\ material}} \left[ \frac{\theta}{(1-\theta)} (\mathbf{1}+a) + \frac{\mathbf{1}}{\rho\_{\text{skeleton}}} \right] / V\_{\text{mixed\ material}} - \mathbf{1} \\ &= \rho\_{\text{mixed}} (\mathbf{1}-\theta) \left[ \frac{\theta}{(\mathbf{1}-\theta)} (\mathbf{1}+a) + \frac{\mathbf{1}}{\rho\_{\text{skeleton}}} \right] - \mathbf{1} \end{aligned} \tag{12}$$

Because the skeleton grain ratio is *Gskeleton* = *ρskeleton*/*ρw*, from formula (12),

$$\eta\_{\text{mixed material}} = \rho\_{\text{mixed}} (\mathbf{1} - \theta) \left[ \frac{\frac{\theta}{(1-\theta)} (\mathbf{1} + a)}{\rho\_{\text{filter}}} + \frac{\mathbf{1}}{\mathbf{G}\_{\text{selection}} \rho\_w} \right] - \mathbf{1} \tag{13}$$

Based on the definition of a mixed material skeleton pore, *e* = *Vresidual pore*/*VS*,

$$\rho\_{\text{mixed}} = \frac{m\_{\text{skeleton}} + m\_{\text{mixed material}}}{V\_{\text{skeleton}} + V\_{\text{mixed material}} + V\_{\text{residual}}\,\text{por}} = \frac{m\_{\text{skeleton}} + m\_{\text{mixed material}}}{(1 + e)(V\_{\text{skeleton}} + V\_{\text{mixed material}})} \tag{14}$$

and employing the skeleton grain ratio,

$$\rho\_{\text{fill}} = \frac{\theta}{\frac{1}{(1+\mathfrak{e})\rho\_{\text{mixed}}} - \frac{(1-\theta)}{(\overline{\mathcal{G}\_{\text{kl;down}}\rho\_{\text{av}}})}} \tag{15}$$

Based on the definition of the porosity ratio,

$$e = \frac{d\_s \rho\_w (1 + 0.001w)}{\rho} - 1 = \frac{d\_s \rho\_w (1 + 0.001w)}{\rho\_{\text{mixed}}} - 1\tag{16}$$

Based on formulas (15) and (16),

$$\rho\_{\text{fill}} = \frac{\theta}{\frac{1}{(1+\epsilon)\rho\_{\text{mid}}} - \frac{(1-\theta)}{(\text{G}\_{\text{ideal}}\rho\_w)}} = \frac{\theta}{\frac{1}{d\_l\rho\_w(1+0.001w)} \cdot (1-\theta)/(\text{G}\_{\text{skeleton}}\rho\_w)}\tag{17}$$

Based on formulas (15) and (17),

$$\eta\_{\text{mixed\ material}} = \rho\_{\text{mixed}}(\mathbf{1} - \theta) \left[ \frac{\frac{\theta}{(\mathbf{1} - \theta)}(\mathbf{1} + a)}{\rho\_{\text{filter}}} + \frac{\mathbf{1}}{\mathbf{G}\_{\text{selection}} \rho\_w} \right] - \mathbf{1} \tag{18}$$

$$= \left[ \frac{\rho\_{\text{mixed}}(\mathbf{1} + a)}{d\_i \rho\_w (\mathbf{1} + \mathbf{0}.\mathbf{0}\mathbf{1}w)} \cdot \frac{\rho\_{\text{mixed}}(\mathbf{1} - \theta)a}{\mathbf{G}\_{\text{selection}} \rho\_w} \right] - \mathbf{1}$$

Formula (18) is simplified to

*ηmixed material* ¼

From formulas (7) and (8),

*Composition of the mixed material.*

*ηmixed material* ¼ *mmixed material*

Based on formulas (6) and (10),

Because

**106**

**Figure 11.**

Δ*Vmixed material πR*<sup>2</sup> *Vmixed material πR*<sup>2</sup>

þ

*Vmixed material*

" #

*<sup>ρ</sup>mixed* <sup>¼</sup> *mskeleton* <sup>þ</sup> *mmixed material*

1 *ρskeleton*

*θ* ð Þ <sup>1</sup>�*<sup>θ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup> ρfiller*

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

<sup>¼</sup> <sup>Δ</sup>*Vmixed material Vmixed material*

*ηfiller* ¼ *α* (8)

*=Vmixed material* � 1 (9)

(7)

(10)

$$
\eta\_{\text{mixed\\_material}} = \eta(\rho\_{\text{mixed}}, \theta, \alpha, d\_s, \rho\_w, w, G\_{\text{selection}}) \tag{19}
$$

Based on formula (19) and an analysis of the situation wherein the microheave filler is completely filled, it was found that mixed material macroscopic expansion is closely related to mixed material compactness *ηmixed material*, filler content *θ*, filler expansion rate *α*, mixed material ratio *ds*, water compactness *ρw*, microheave filler water content *w* and skeleton ratio *Gskeleton*. Actual frost heave rates are greater than the calculated values.

#### **5.2 Microheave filler partial filling**

To further investigate the mixed material frost heave rule, microscopic analysis is required to understand the interactions between pore filler and microheave filler. During filler expansion, mixed material skeleton pores cannot be filled completely. Microheave filler specimens would expand under the side constraint condition. During expansion, the filler would experience plastic deformation and fill skeleton pores, which would result in microheave filler specimen macroscopic elevation. The manifestation of core filler stress is very complex. First, filler grains are not uniformly distributed in all skeleton pores of mixed materials. When stresses on fine aggregates are small, coarse aggregates are not affected, and coarse aggregate skeletons experience no deformation; coarse aggregate and fine aggregate contact surfaces are equivalent to the displacement boundary condition. As the filler expands and fills the remaining part of the skeleton pores, although the fine aggregate stresses are significant, because the expanded filler does not fill all the skeleton pores in the mixed material, the mixed material skeleton grains are not lifted, and the mixed material skeleton does not swell. To summarize, fine aggregates are not completely surrounded by coarse aggregates, pores are interconnected, fine aggregates either expand and deform in large pores or are squeezed to adjacent pores. Therefore, as shown in **Figure 12**, stress state cannot be described via simple mechanical models.

Based on Griffith micro-fracture theory and the minimum energy principle, a calculation formula was deduced via mathematical and mechanical analysis to undertake a quantitative analysis of the effects of rock and soil rheological properties for engineering. Filler was pushed into a cylinder using a porous piston. An analogy from physics was applied, and the filler expansion stress status was compared to the mathematical physics model in **Figure 13**.

Each pore in the microheave filler skeleton was treated as a container, as shown in **Figure 13**. The container top plate was porous, and more holes implied greater skeleton porosity.

The filling rate reflects the filling degree, which is defined as the ratio of filler volume to skeleton pore volume. Assuming that the mixed material skeleton grain pore filling rate by filler grains is *β*,

$$\beta = \frac{V\_{filter}}{V\_{selection\ pore}}\tag{20}$$

For an initial filling rate *β0*, employing formula (20) yields

$$V\_{skeleton\ power} = \frac{V\_{filter}}{\beta} \tag{21}$$

<sup>Δ</sup>*Vmixed material* <sup>¼</sup> <sup>Δ</sup>*Vskeleton pore* <sup>¼</sup> <sup>Δ</sup>*Vfiller*

**Figure 12.**

**Figure 13.**

**109**

*Mixed material partial filling.*

*Skeleton pore filled with filler.*

*Frost Heave Deformation Analysis Model for Microheave Filler*

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

When the filler grains expanded, the porous plate moved upward, which increased the macroscopic volume. In addition, some fine grains were extruded via

*β* þ

*Vfiller*

*<sup>β</sup>* � *Vfiller β*0

(22)

When filler fills skeleton pores, the mixed material skeleton grain volume does not change during expansion. Mixed material macroscopic volume changes can only be achieved by changing the skeleton pores, as shown by formula (22).

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

**Figure 12.** *Skeleton pore filled with filler.*

*ηmixed material* ¼ *η ρmixed; θ; α; ds* ð Þ *; ρw; w; Gskeleton* (19)

Based on formula (19) and an analysis of the situation wherein the microheave filler is completely filled, it was found that mixed material macroscopic expansion is closely related to mixed material compactness *ηmixed material*, filler content *θ*, filler expansion rate *α*, mixed material ratio *ds*, water compactness *ρw*, microheave filler water content *w* and skeleton ratio *Gskeleton*. Actual frost heave rates are greater than

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

To further investigate the mixed material frost heave rule, microscopic analysis is required to understand the interactions between pore filler and microheave filler. During filler expansion, mixed material skeleton pores cannot be filled completely. Microheave filler specimens would expand under the side constraint condition. During expansion, the filler would experience plastic deformation and fill skeleton pores, which would result in microheave filler specimen macroscopic elevation. The manifestation of core filler stress is very complex. First, filler grains are not uniformly distributed in all skeleton pores of mixed materials. When stresses on fine aggregates are small, coarse aggregates are not affected, and coarse aggregate skeletons experience no deformation; coarse aggregate and fine aggregate contact surfaces are equivalent to the displacement boundary condition. As the filler expands and fills the remaining part of the skeleton pores, although the fine aggregate stresses are significant, because the expanded filler does not fill all the skeleton pores in the mixed material, the mixed material skeleton grains are not lifted, and the mixed material

skeleton does not swell. To summarize, fine aggregates are not completely surrounded by coarse aggregates, pores are interconnected, fine aggregates either expand and deform in large pores or are squeezed to adjacent pores. Therefore, as shown in **Figure 12**, stress state cannot be described via simple mechanical models. Based on Griffith micro-fracture theory and the minimum energy principle, a calculation formula was deduced via mathematical and mechanical analysis to undertake a quantitative analysis of the effects of rock and soil rheological properties for engineering. Filler was pushed into a cylinder using a porous piston. An analogy from physics was applied, and the filler expansion stress status was com-

Each pore in the microheave filler skeleton was treated as a container, as shown in **Figure 13**. The container top plate was porous, and more holes implied greater

The filling rate reflects the filling degree, which is defined as the ratio of filler volume to skeleton pore volume. Assuming that the mixed material skeleton grain

> *<sup>β</sup>* <sup>¼</sup> *Vfiller Vskeleton pore*

*Vskeleton pore* <sup>¼</sup> *Vfiller*

When filler fills skeleton pores, the mixed material skeleton grain volume does not change during expansion. Mixed material macroscopic volume changes can only

For an initial filling rate *β0*, employing formula (20) yields

be achieved by changing the skeleton pores, as shown by formula (22).

(20)

*<sup>β</sup>* (21)

pared to the mathematical physics model in **Figure 13**.

the calculated values.

skeleton porosity.

**108**

pore filling rate by filler grains is *β*,

**5.2 Microheave filler partial filling**

**Figure 13.** *Mixed material partial filling.*

$$
\Delta V\_{\text{mixed material}} = \Delta V\_{\text{selection pore}} = \frac{\Delta V\_{\text{filter}}}{\beta} + \frac{V\_{\text{filter}}}{\beta} - \frac{V\_{\text{filter}}}{\beta\_0} \tag{22}
$$

When the filler grains expanded, the porous plate moved upward, which increased the macroscopic volume. In addition, some fine grains were extruded via the top plate pore, and plastic deformation consumed energy, as shown by formula (23).

$$
\Delta V\_{filter} = \Delta V\_{mixed\ material} + \Delta V\_{extrule} \tag{23}
$$

Δ*H* ¼ Δ*Eelastic potential energy* þ *Wwork* þ *Wplastic deformation energy consumption*

1 1‐*β*

*<sup>β</sup>* � *Vmixed material*

According to the principle of minimum energy, the total internal energy of a closed system with steady volume, external parameters and entropy will tend to decrease. When a balance state is reached, the overall internal energy reaches the minimum level. *β* should result in a minimum Δ*H*; the range of *β* is *β*<sup>0</sup> ≤ *β* ≤ 1.

*∂*Δ*H*

<sup>þ</sup> *χ α* � *<sup>α</sup>*

When *α =* 0, *β* = *β*0. This is substituted into the above formula, yielding

� �*β*<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*ab* � <sup>2</sup>*<sup>χ</sup>* � <sup>2</sup>*αχ* <sup>þ</sup> <sup>2</sup>*<sup>b</sup> <sup>β</sup>* <sup>þ</sup> <sup>ð</sup>*αχ* � *<sup>b</sup>* � *ab* <sup>þ</sup> *<sup>χ</sup>*Þ ¼ 0 (33)

*<sup>β</sup>* <sup>¼</sup> *<sup>A</sup>* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>B</sup>* � *<sup>C</sup>* � *<sup>D</sup>* <sup>p</sup>

*A* ¼ �ð Þ 2*ab* � 2*χ* � 2*αχ* þ 2*b , <sup>B</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*ab* � <sup>2</sup>*<sup>χ</sup>* � <sup>2</sup>*αχ* <sup>þ</sup> <sup>2</sup>*<sup>b</sup>* <sup>2</sup>

� �*,*

� �*:*

*<sup>C</sup>* <sup>¼</sup> <sup>4</sup> *αχ* � *<sup>b</sup>* � *<sup>α</sup><sup>b</sup>* <sup>þ</sup> *<sup>χ</sup>*

*D* ¼ ð Þ *αχ* � *b* � *αb* þ *χ , <sup>E</sup>* <sup>¼</sup> <sup>2</sup> *αχ* � *<sup>b</sup>* � *<sup>α</sup><sup>b</sup>* <sup>þ</sup> *<sup>χ</sup>*

Because the range for *β* is *β*<sup>0</sup> ≤ *β* ≤ 1, the solution for the single variable quadratic

It is easy to prove that when *α* ! þ∞, *β* ! 1�. The solution is

The filler grain volume expansion rate is *α =* Δ*V*mixed material/*V*mixed material, and

*β* � 1 *β* þ 1 *β*0

� � 1

*β*0

� � 1

*<sup>β</sup>* � *Vmixed material*

*β* þ

*<sup>∂</sup><sup>β</sup>* <sup>¼</sup> <sup>0</sup> (31)

*<sup>E</sup>* (34)

(35)

*,*

*β*0

*β*0

*Vmixed material β*0

ð Þ <sup>1</sup> � *<sup>β</sup>* <sup>2</sup> <sup>¼</sup> 0 (32)

(29)

1‐*β* (30)

¼ Δ*Vmixed materialpv* þ *χKPΔVextrude*

*Frost Heave Deformation Analysis Model for Microheave Filler*

<sup>þ</sup> *<sup>χ</sup>KP* <sup>Δ</sup>*Vmixed material* � <sup>Δ</sup>*Vmixed material*

*Vmixed material*

� �*pv*

Based on formulas (22), (24) and (29),

*β* þ

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

The prerequisite for minimum Δ*H* is

*b* = *pv*/*KP*; then, based on (31),

*χ* � *b* � � *α*

*χ* � *b*=0, where *χ* ¼ 1 � *β*<sup>0</sup> ð Þ*b*.

*β*0

*β*<sup>2</sup> þ

1 *β*2 � �

1 1 � *β*

*αχ* � *<sup>b</sup>* � *<sup>α</sup><sup>b</sup>* <sup>þ</sup> *<sup>χ</sup>*

equation (33) is

where

**111**

1 1�*β*<sup>0</sup>

<sup>Δ</sup>*<sup>H</sup>* <sup>¼</sup> <sup>Δ</sup>*Vmixed material*

Based on formulas (22) and (23),

$$
\Delta V\_{extrule} = \Delta V\_{filter} - \frac{\Delta V\_{filter}}{\beta} - \frac{V\_{filter}}{\beta} + \frac{V\_{filter}}{\beta\_0} \tag{24}
$$

Because filler grains are in a plastic state, their elastic potential energies cannot be increased. Therefore, Δ*Eelastic potential energy* = 0.

The mixed material expands by Δ*Vall*, the upper load is *N*, and the pressure is *pv* = *N*/*S*2. Assume that the piston area is *S*=*S*<sup>1</sup> + *S*2, where *S*<sup>1</sup> is the piston area with pores, and *S*<sup>2</sup> is the piston area without pores. Therefore, the work during expansion is

$$W\_{work} = \Delta V\_{all} \, p\_v \tag{25}$$

Note: Δ*Vall pv* has units of energy, whereas Δ*Vall N* does not have units of energy.

The extrusion volume is calculated via formula (24). The corresponding soil develops plastic deformation and dissipates energy. Based on a general expression of the plastic strain flow rule under the principle of minimum energy consumption, the plastic deformation energy consumption rate is

$$
\eta = \int\_v \rho \rho dv \tag{26}
$$

Where *ρφ* = *σdε p* . The energy dissipation process only occurs when the yield criterion is satisfied. In this paper, plastic energy is the product of the stress friction coefficient and volume strain. That is, *Wplastic* deformation energy consumption is proportional to the soil internal friction coefficient *K*, soil stress *P* and soil extrusion volume Δ*Vextrude*. In addition, when *β* is larger and the area of pores in the porous plate is smaller, it is more difficult to extrude soil, and energy consumption is higher. Therefore, it should be positively correlated with 1/(1-*β*). Plastic energy consumption is described by formula (27):

$$\mathcal{W}\_{\text{plastic deformation energy consumption}} = \frac{\chi \text{KP} \Delta V\_{\text{extrude}}}{(1 - \beta)} \tag{27}$$

where *K* is a dimensionless variable, *P* is stress, *χ* is a constant, and *χ* = (1-*β0*)*b*. The solution for *P* now follows.

A microelement under a porous plate hole to be extruded is in a plastic flow state, and *σ*<sup>3</sup> = 0. *σ*<sup>1</sup> = *σ*<sup>2</sup> > 0. Based on the Mohr-Coulomb criterion, *σ*<sup>1</sup> = *σ*<sup>2</sup> = 2*c*cos*φ*/ (1-sin*φ*).

Therefore,

$$P = \frac{\sigma\_1 + \sigma\_2 + \sigma\_3}{3} = \frac{4}{3}c \frac{\cos \rho}{1 - \sin \rho} \tag{28}$$

*K* represents soil friction, and *K* = tan*φ*. In this process, the expansion potential energy is *Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

Δ*H* ¼ Δ*Eelastic potential energy* þ *Wwork* þ *Wplastic deformation energy consumption*

$$
\dot{\rho} = \Delta V\_{\text{mixed material}} p\_v + \chi \text{KP} \Delta V\_{\text{extrude}} \frac{\mathbf{1}}{\mathbf{1} \cdot \boldsymbol{\beta}} \tag{29}
$$

Based on formulas (22), (24) and (29),

the top plate pore, and plastic deformation consumed energy, as shown by

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

<sup>Δ</sup>*Vextrude* <sup>¼</sup> <sup>Δ</sup>*Vfiller* � <sup>Δ</sup>*Vfiller*

Δ*Vfiller* ¼ Δ*Vmixed material* þ Δ*Vextrude* (23)

*β* þ

*Wwork* ¼ Δ*Vall pv* (25)

*ρφdv* (26)

ð Þ <sup>1</sup> � *<sup>β</sup>* (27)

(28)

*Vfiller β*0

(24)

*<sup>β</sup>* � *Vfiller*

Because filler grains are in a plastic state, their elastic potential energies cannot

The mixed material expands by Δ*Vall*, the upper load is *N*, and the pressure is *pv* = *N*/*S*2. Assume that the piston area is *S*=*S*<sup>1</sup> + *S*2, where *S*<sup>1</sup> is the piston area with pores, and *S*<sup>2</sup> is the piston area without pores. Therefore, the work during

Note: Δ*Vall pv* has units of energy, whereas Δ*Vall N* does not have units of energy. The extrusion volume is calculated via formula (24). The corresponding soil develops plastic deformation and dissipates energy. Based on a general expression of the plastic strain flow rule under the principle of minimum energy consumption,

> *ψ* ¼ ð

> > *v*

criterion is satisfied. In this paper, plastic energy is the product of the stress friction coefficient and volume strain. That is, *Wplastic* deformation energy consumption is proportional to the soil internal friction coefficient *K*, soil stress *P* and soil extrusion volume Δ*Vextrude*. In addition, when *β* is larger and the area of pores in the porous plate is smaller, it is more difficult to extrude soil, and energy consumption is higher. Therefore, it should be positively correlated with 1/(1-*β*). Plastic energy

*Wplastic deformation energy consumption* <sup>¼</sup> *<sup>χ</sup>KP*Δ*Vextrude*

where *K* is a dimensionless variable, *P* is stress, *χ* is a constant, and *χ* = (1-*β0*)*b*.

A microelement under a porous plate hole to be extruded is in a plastic flow state, and *σ*<sup>3</sup> = 0. *σ*<sup>1</sup> = *σ*<sup>2</sup> > 0. Based on the Mohr-Coulomb criterion, *σ*<sup>1</sup> = *σ*<sup>2</sup> = 2*c*cos*φ*/

<sup>3</sup> <sup>¼</sup> <sup>4</sup>

3 *c*

cos *φ* 1 � sin *φ*

*<sup>P</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>3</sup>

. The energy dissipation process only occurs when the yield

formula (23).

expansion is

Where *ρφ* = *σdε*

(1-sin*φ*).

**110**

Therefore,

*p*

consumption is described by formula (27):

The solution for *P* now follows.

*K* represents soil friction, and *K* = tan*φ*.

In this process, the expansion potential energy is

Based on formulas (22) and (23),

be increased. Therefore, Δ*Eelastic potential energy* = 0.

the plastic deformation energy consumption rate is

$$\begin{split} \Delta H &= \left(\frac{\Delta V\_{\text{mixed material}}}{\beta} + \frac{V\_{\text{mixed material}}}{\beta} - \frac{V\_{\text{mixed material}}}{\beta\_0}\right) p\_v \\ &+ \chi \text{KP}\left(\Delta V\_{\text{mixed material}} - \frac{\Delta V\_{\text{mixed material}}}{\beta} - \frac{V\_{\text{mixed material}}}{\beta} + \frac{V\_{\text{mixed material}}}{\beta\_0}\right) \frac{\mathbf{1}}{\mathbf{1} \cdot \boldsymbol{\beta}} \end{split} \tag{30}$$

According to the principle of minimum energy, the total internal energy of a closed system with steady volume, external parameters and entropy will tend to decrease. When a balance state is reached, the overall internal energy reaches the minimum level. *β* should result in a minimum Δ*H*; the range of *β* is *β*<sup>0</sup> ≤ *β* ≤ 1.

The prerequisite for minimum Δ*H* is

$$\frac{\partial \Delta H}{\partial \beta} = \mathbf{0} \tag{31}$$

The filler grain volume expansion rate is *α =* Δ*V*mixed material/*V*mixed material, and *b* = *pv*/*KP*; then, based on (31),

$$
\chi \left( \frac{1}{1 - \beta} \chi - b \right) \left( \frac{a}{\beta^2} + \frac{1}{\beta^2} \right) + \chi \left( a - \frac{a}{\beta} - \frac{1}{\beta} + \frac{1}{\beta\_0} \right) \frac{1}{\left( 1 - \beta \right)^2} = 0 \tag{32}
$$

When *α =* 0, *β* = *β*0. This is substituted into the above formula, yielding 1 1�*β*<sup>0</sup> *χ* � *b*=0, where *χ* ¼ 1 � *β*<sup>0</sup> ð Þ*b*.

It is easy to prove that when *α* ! þ∞, *β* ! 1�. The solution is

$$\left(a\chi - b - ab + \frac{\chi}{\beta\_0}\right)\beta^2 + (2ab - 2\chi - 2a\chi + 2b)\beta + (a\chi - b - ab + \chi) = 0 \tag{33}$$

Because the range for *β* is *β*<sup>0</sup> ≤ *β* ≤ 1, the solution for the single variable quadratic equation (33) is

$$
\beta = \frac{A \pm \sqrt{B - C \cdot D}}{E} \tag{34}
$$

where

$$\begin{aligned} A &= -(2ab - 2\chi - 2a\chi + 2b), \\ B &= (2ab - 2\chi - 2a\chi + 2b)^2, \\ C &= 4\left(a\chi - b - ab + \frac{\chi}{\beta\_0}\right), \\ D &= (a\chi - b - ab + \chi), \\ E &= 2\left(a\chi - b - ab + \frac{\chi}{\beta\_0}\right). \end{aligned} \tag{35}$$

Based on formula (34), this is simplified to

$$
\beta = \beta(a, b, \chi, \beta\_0) \tag{36}
$$

<sup>Δ</sup>*Vmixed material* <sup>¼</sup> <sup>Δ</sup>*Vfiller* � *<sup>α</sup>* � *<sup>α</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

*Frost Heave Deformation Analysis Model for Microheave Filler*

where

*<sup>β</sup>* � <sup>1</sup> *<sup>β</sup>* <sup>þ</sup> <sup>1</sup> *β*0

!

*<sup>β</sup>* � <sup>1</sup> *<sup>β</sup>* <sup>þ</sup> <sup>1</sup> *β*0

<sup>Δ</sup>*Vmixed material* <sup>¼</sup> <sup>Δ</sup>*Vfiller* � *M Vskeleton pore* � *Vfiller* � � (44)

*ρkeleton*

þ

*M ρskeleton*

þ

þ

3 7 <sup>5</sup> � *<sup>M</sup>*

*M ρskeleton*

*M Gskeletonρ<sup>w</sup>*

3 7 <sup>5</sup> � *<sup>M</sup>*

> 3 7 <sup>5</sup> � *<sup>M</sup>*

*=Vmixed material* � *M*

*M ρskeleton*

þ

ð Þ *M* þ *α*

ð Þ *M* þ *α*

*ρfiller*

*ρfiller*

*M <sup>ρ</sup>skeleton* " #*=Vmixed material* � *<sup>M</sup>* (47)

1

*Vskeleton pore* � *Vfiller* � � (42)

*<sup>β</sup>* � <sup>1</sup> (43)

� *MVmixed material*

(45)

ð46Þ

(48)

(49)

1 *<sup>β</sup>* � 1

*<sup>M</sup>* <sup>¼</sup> *<sup>α</sup>* � *<sup>α</sup>*

The above formula is quantified. Based on formulas (4) and (44),

ð Þþ *<sup>M</sup>* <sup>þ</sup> *<sup>α</sup> <sup>M</sup> mkeleton*

*θ* ð Þ <sup>1</sup>�*<sup>θ</sup>* ð Þ *<sup>M</sup>* <sup>þ</sup> *<sup>α</sup> ρfiller*

þ

ð Þ *M* þ *α ρfiller*

The microheave filler frost heave model is simplified to

Δ*Vmixed material* ¼ Δ*Vfiller* � *MVresidual pore* <sup>¼</sup> *mfiller ρfiller*

Based on formulas (3), (45) is rearranged to

Based on formulas (7), (8) and (46),

*Vmixed material*

Substitution of formulas (11) into (47) yields

¼ *ρmixed*ð Þ 1 � *θ*

*ηmixed material* ¼ *ρmixed*ð Þ 1 � *θ*

**113**

¼ *ρmixed*ð Þ 1 � *θ*

¼ *mskeleton*

*θ*

ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ *<sup>M</sup>* <sup>þ</sup> *<sup>α</sup> ρfiller*

> *θ* 1 � *θ*

Based on the skeleton grain ratio definition, (49) is rearranged as

2 6 4

2 6 4

*θ* 1 � *θ*

*θ* 1 � *θ*

2 6 4

*<sup>η</sup>mixed material* <sup>¼</sup> <sup>Δ</sup>*Vmixed material*

*ηmixed material* ¼ *mskeleton*

where *b* ¼ *pv=KP*, *pv* ¼ *N=S* ¼ *N*ð Þ 1 þ *e* , and *χ* ¼ 1 � *β*<sup>0</sup> ð Þ*b* . Formula (36) is further optimized:

$$
\beta = \beta(\mathbf{N}, \alpha, \mathbf{C}, \boldsymbol{\varrho}, \boldsymbol{\beta}\_0) \tag{37}
$$

The formula for calculating the partial filling model expansion filling rate shows that the expansion filling rate is closely related to parameters such as the overlying load, filler cohesion, friction angle and initial fill rate.

The relations between these physical variables are

$$\sigma\_{skelenton0} = \frac{V\_{skelenton\ power}}{V\_{skelenton\ solid}}\tag{38}$$

$$
\beta\_0 \mathbf{V}\_{skeleton\ por} = \mathbf{V}\_{\text{filter}} \tag{39}
$$

During expansion,

$$\begin{split} \varepsilon\_{sklepton} &= \frac{V\_{skelectron\ pore} + \Delta V\_{skelectron\ pore}}{V\_{skelectron\ solid}} = \varepsilon\_{skelectron0} + \frac{\Delta V\_{skelectron\ pore}}{V\_{skelectron\ solid}} \\ &= \varepsilon\_0 + \frac{\Delta V\_{filter}}{V\_{skelectron\ solid}} \left( \frac{a}{\beta} + \frac{1}{\beta} - \frac{1}{\beta\_0} \right) \end{split} \tag{40}$$

As the skeleton grain pore increases, Δ*Vskeleton pore* > 0, *eskeleton* increases accordingly.

Based on formula (40), if there is only an elevation effect, *β*<sup>0</sup> = *β*, the ratio of Δ*Vskeleton pore* versus Δ*Vfiller* does not change, and the following always holds:

$$\Delta V\_{skelton\ pore} + \Delta V\_{skelton\ pore} = \frac{\left(V\_{filter} + \Delta V\_{filter}\right)}{\beta\_0} \tag{41}$$

Ease of extrusion is directly related to *β*. In formula (40), if Δ*Vfiller* is fixed and only *β* increases, then Δ*Vskeleton pore* decreases; i.e., small extrusion holes and lower elevations make extrusion difficult.

During filler grain expansion, elevation is calculated via formula (31), and extrusion is calculated via formula (32). The ratio of the two is calculated by substituting *β* into the two formulae.

#### **5.3 Creation of a microheave filler frost heave model**

In an actual project, when the filler is expanding, microheave filler skeleton pores cannot be filled completely. Therefore, the microheave filler frost heave complete filling model should be modified. The microheave filler complete filling theoretical model and partial filling model are combined, and interactions between the filling and elevation effects are considered. The model is created using the principle of minimum energy. For complete filling, the microheave filler frost heave model is described by formula (3). For partial filling, the microheave filler model is described by formula (23).

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

$$
\Delta V\_{\text{mixed\ material}} = \Delta V\_{\text{fill}r} - \left(\frac{a - \frac{a}{\beta} - \frac{1}{\beta} + \frac{1}{\beta\_0}}{\frac{1}{\beta} - \mathbf{1}}\right) \left(V\_{\text{selection\ power}} - V\_{\text{filter}}\right) \tag{42}
$$

where

Based on formula (34), this is simplified to

load, filler cohesion, friction angle and initial fill rate. The relations between these physical variables are

*eskeleton* <sup>¼</sup> *Vskeleton pore* <sup>þ</sup> <sup>Δ</sup>*Vskeleton pore*

¼ *e*<sup>0</sup> þ

elevations make extrusion difficult.

substituting *β* into the two formulae.

described by formula (23).

**112**

*Vskeleton* solid

*α β* þ 1 *<sup>β</sup>* � <sup>1</sup> *β*0

As the skeleton grain pore increases, Δ*Vskeleton pore* > 0, *eskeleton* increases

Δ*Vskeleton pore* versus Δ*Vfiller* does not change, and the following always holds:

Based on formula (40), if there is only an elevation effect, *β*<sup>0</sup> = *β*, the ratio of

*Vskeleton pore* <sup>þ</sup> <sup>Δ</sup>*Vskeleton pore* <sup>¼</sup> *Vfiller* <sup>þ</sup> <sup>Δ</sup>*Vfiller*

During filler grain expansion, elevation is calculated via formula (31), and extrusion is calculated via formula (32). The ratio of the two is calculated by

In an actual project, when the filler is expanding, microheave filler skeleton pores cannot be filled completely. Therefore, the microheave filler frost heave complete filling model should be modified. The microheave filler complete filling theoretical model and partial filling model are combined, and interactions between the filling and elevation effects are considered. The model is created using the principle of minimum energy. For complete filling, the microheave filler frost heave model is described by formula (3). For partial filling, the microheave filler model is

Ease of extrusion is directly related to *β*. In formula (40), if Δ*Vfiller* is fixed and only *β* increases, then Δ*Vskeleton pore* decreases; i.e., small extrusion holes and lower

Δ*Vfiller Vskeleton* solid

**5.3 Creation of a microheave filler frost heave model**

Formula (36) is further optimized:

During expansion,

accordingly.

where *b* ¼ *pv=KP*, *pv* ¼ *N=S* ¼ *N*ð Þ 1 þ *e* , and *χ* ¼ 1 � *β*<sup>0</sup> ð Þ*b* .

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

The formula for calculating the partial filling model expansion filling rate shows that the expansion filling rate is closely related to parameters such as the overlying

*eskeleton*<sup>0</sup> <sup>¼</sup> *Vskeleton pore*

*Vskeleton* solid

¼ *eskeleton*<sup>0</sup> þ

*β* ¼ *β α; b; χ; β*<sup>0</sup> ð Þ (36)

*β* ¼ *β N; α;C; φ; β*<sup>0</sup> ð Þ (37)

*β*0*Vskeleton pore* ¼ *Vfiller* (39)

(40)

 *β*0

Δ*Vskeleton pore Vskeleton* solid

(38)

(41)

$$M = \frac{a - \frac{a}{\beta} - \frac{1}{\beta} + \frac{1}{\beta\_0}}{\frac{1}{\beta} - 1} \tag{43}$$

The microheave filler frost heave model is simplified to

$$
\Delta V\_{\text{mixed material}} = \Delta V\_{\text{filter}} - M \left( V\_{\text{selection power}} - V\_{\text{filter}} \right) \tag{44}
$$

The above formula is quantified. Based on formulas (4) and (44),

$$\begin{split} \Delta V\_{\text{mixed material}} &= \Delta V\_{\text{filter}} - MV\_{\text{residual}} \,\text{port} \\ &= \frac{m\_{\text{filter}}}{\rho\_{\text{filter}}} (\mathcal{M} + \alpha) + \mathcal{M} \frac{m\_{\text{selection}}}{\rho\_{\text{selection}}} - \mathcal{M} V\_{\text{mixed material}} \end{split} \tag{45}$$

Based on formulas (3), (45) is rearranged to

$$
\Delta V\_{\% \otimes \text{\u0t}} = m\_{\text{\u0t}} \left[ \frac{\frac{\partial}{1-\Theta} (M+\alpha)}{\rho\_{\text{\u0t} \otimes \text{\u0t}}} + \frac{M}{\rho\_{\text{\u0t} \otimes \text{\u0}}} \right] - M V\_{\text{\u0t} \otimes \text{\u0t}} \tag{46}$$

Based on formulas (7), (8) and (46),

$$\eta\_{\text{mixed material}} = \frac{\Delta V\_{\text{mixed material}}}{V\_{\text{mixed material}}} = m\_{\text{skeleton}} \left[ \frac{\frac{\theta}{(1-\theta)}(M+\alpha)}{\rho\_{\text{filter}}} + \frac{M}{\rho\_{\text{skeleton}}} \right] / V\_{\text{mixed material}} - M \tag{47}$$

Substitution of formulas (11) into (47) yields

$$\begin{aligned} \eta\_{\text{mixed\ material}} &= m\_{\text{skeleton}} \left[ \frac{\theta}{(1-\theta)} (M+a) + \frac{M}{\rho\_{\text{skeleton}}} \right] / V\_{\text{mixed\ material}} - M \\ &= \rho\_{\text{mixed}} (1-\theta) \left[ \frac{\theta}{1-\theta} (M+a) + \frac{M}{\rho\_{\text{skeleton}}} \right] - M \end{aligned} \tag{48}$$

Based on the skeleton grain ratio definition, (49) is rearranged as

$$\begin{split} \eta\_{\text{mixed material}} &= \rho\_{\text{mixed}} (\mathbf{1} - \theta) \left[ \frac{\frac{\theta}{\mathbf{1} - \theta} (M + \alpha)}{\rho\_{\text{filter}}} + \frac{M}{\rho\_{\text{selection}}} \right] - M \\ &= \rho\_{\text{mixed}} (\mathbf{1} - \theta) \left[ \frac{\frac{\theta}{\mathbf{1} - \theta} (M + \alpha)}{\rho\_{\text{filter}}} + \frac{M}{G\_{\text{skeleton}} \rho\_w} \right] - M \end{split} \tag{49}$$

Based on formula (17), the left side is Δ*Vall = ηmixed hS*. Based on the complete filling model, *hS* is 1. Rearranging formula (50) yields

$$\begin{aligned} \eta\_{\text{mixed\ material}} &= \rho\_{\text{mixed}} (1 - \theta) \left[ \frac{\theta}{1 - \theta} (M + a) + \frac{M}{\rho\_{\text{global}}} \right] - M \\ &= \left[ \frac{\rho\_{\text{mixed}} (M + a)}{d\_i \rho\_{\text{filter}} (1 + 0.01 w)} + \frac{\rho\_{\text{mixed}} (1 - \theta) a}{G\_{\text{skelton}} \rho\_w} \right] - M \end{aligned} \tag{50}$$

where *M* is given by formula (43), and *β* in *M* is given by formula (35).

To determine *β*, *b* = *pv*/*KP* is used, where K represents soil friction, *K* = tan*φ*, and *P* is determined using formula (33).

The pressure is *pv* = *N*/*S*=*N*(1 + *e*), and *χ* is a constant, where *χ* = (1-*β*0)*b*. Formula (51) is simplified to

$$\eta\_{\text{mixed\ material}} = \eta(\mathcal{N}, \rho, \mathcal{C}, a, \beta\_0, d\_{\text{s}}, \rho\_w, w, \rho\_{\text{mixed}}, \theta, \mathcal{G}\_{\text{skeleton}}) \tag{51}$$

The parameters in relation formula (51) were obtained via the indoor filler frost heave test, and the simple physical and mechanical test was used for the microheave filler.

Based on the microheave filler frost heave model, microheave filler frost heave is closely related to the specimen overlying load, filler plastic deformation friction angle, filler cohesion, filler frost heave rate, mixed material skeleton pore filling rate by the initial filler, mixed material ratio, water density, mixed material water content, mixed material density, filler content and mixed material skeleton ratio.

#### **6. Model verification**

Based on the model presented in this paper, the filler frost heave rates for various filler filling rates are shown in **Figure 14**, and the filler frost heave rates for various water contents are shown in **Figure 15**. These results were then compared with the test results presented in this paper.

**Figures 14** and **15** show that the model simulation results essentially matched the test results. However, the specimens in the test and simulation differed. A silt filler frost heave rate test under an overlying load of *N*<sup>1</sup> = 60 kPa was performed. Other parameters, including the model parameters, are listed in **Table 5**.

The filler frost heave rate in the lab is *ηmixed material* = 0.123 mm/ 150 mm = 0.082%.

The simulation showed that when the water content was 9%, the frost heave rate was approximately 0.22%.

A comparison of the calculated and experimental filler frost heave rates showed a difference of only 0.06%, which indicates that the microheave filler frost heave computational model derived in this paper is viable for frost heave rate calculations.

condition, on the subgrade frost heave rate with or without external constraints (overlying train loads) were investigated via an indoor frost heave characteristics test. On that basis, models for complete and partial filling microheave filler frost heave were created. The validity of the theoretical microheave filler frost heave rate forecast model was demonstrated via indoor testing. The detailed conclusions are as

1. The filler grading frost heave test showed that when the gravel or finer aggregate was reduced, the filler frost heave rate decreased gradually; however, even for very low levels of fine grain content at low temperatures,

follows.

**115**

**Figure 14.**

*Frost heave rate versus filler content.*

*Frost Heave Deformation Analysis Model for Microheave Filler*

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

#### **7. Conclusions**

To solve problems of seasonal permafrost region subgrade filler microheave in China, the effects of influencing factors, such as the filler content, natural water holding capacity, filler grading, permeability coefficient and water supply

*Frost Heave Deformation Analysis Model for Microheave Filler DOI: http://dx.doi.org/10.5772/intechopen.82575*

Based on formula (17), the left side is Δ*Vall = ηmixed hS*. Based on the complete

ð Þ *M* þ *α*

þ

<sup>þ</sup> *<sup>ρ</sup>mixed*ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup> Gskeletonρ<sup>w</sup>*

*M ρskeleton*

3 7 <sup>5</sup> � *<sup>M</sup>*

� *M*

(50)

*ρfiller*

" #

To determine *β*, *b* = *pv*/*KP* is used, where K represents soil friction, *K* = tan*φ*, and

The parameters in relation formula (51) were obtained via the indoor filler frost heave test, and the simple physical and mechanical test was used for the microheave

Based on the microheave filler frost heave model, microheave filler frost heave is closely related to the specimen overlying load, filler plastic deformation friction angle, filler cohesion, filler frost heave rate, mixed material skeleton pore filling rate by the initial filler, mixed material ratio, water density, mixed material water content, mixed material density, filler content and mixed material skeleton ratio.

Based on the model presented in this paper, the filler frost heave rates for various filler filling rates are shown in **Figure 14**, and the filler frost heave rates for various water contents are shown in **Figure 15**. These results were then compared

**Figures 14** and **15** show that the model simulation results essentially matched the test results. However, the specimens in the test and simulation differed. A silt filler frost heave rate test under an overlying load of *N*<sup>1</sup> = 60 kPa was performed.

The simulation showed that when the water content was 9%, the frost heave rate

A comparison of the calculated and experimental filler frost heave rates showed a difference of only 0.06%, which indicates that the microheave filler frost heave computational model derived in this paper is viable for frost heave rate calculations.

To solve problems of seasonal permafrost region subgrade filler microheave in China, the effects of influencing factors, such as the filler content, natural water holding capacity, filler grading, permeability coefficient and water supply

Other parameters, including the model parameters, are listed in **Table 5**. The filler frost heave rate in the lab is *ηmixed material* = 0.123 mm/

*ηmixed material* ¼ *η N; φ;C; α; β*0*; ds; ρw; w; ρmixed* ð Þ *; θ; Gskeleton* (51)

*θ* 1 � *θ*

where *M* is given by formula (43), and *β* in *M* is given by formula (35).

The pressure is *pv* = *N*/*S*=*N*(1 + *e*), and *χ* is a constant, where *χ* = (1-*β*0)*b*.

2 6 4

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

<sup>¼</sup> *<sup>ρ</sup>mixed*ð Þ *<sup>M</sup>* <sup>þ</sup> *<sup>α</sup> dsρfiller*ð Þ 1 þ 0*:*01*w*

filling model, *hS* is 1. Rearranging formula (50) yields

*ηmixed material* ¼ *ρmixed*ð Þ 1 � *θ*

*P* is determined using formula (33).

Formula (51) is simplified to

**6. Model verification**

150 mm = 0.082%.

**7. Conclusions**

**114**

was approximately 0.22%.

with the test results presented in this paper.

filler.

**Figure 14.** *Frost heave rate versus filler content.*

condition, on the subgrade frost heave rate with or without external constraints (overlying train loads) were investigated via an indoor frost heave characteristics test. On that basis, models for complete and partial filling microheave filler frost heave were created. The validity of the theoretical microheave filler frost heave rate forecast model was demonstrated via indoor testing. The detailed conclusions are as follows.

1. The filler grading frost heave test showed that when the gravel or finer aggregate was reduced, the filler frost heave rate decreased gradually; however, even for very low levels of fine grain content at low temperatures,

the soil still produced a certain level of frost heave. The filler overlying load test showed that an overlying pressure produced a certain level of suppression

2. Microheave filler frost heave increased with filler frost heave; however, when the filler fill rate was under 0.25, as the filler content increased, the changes in filler frost heave were insignificant. This means that at this condition, filler frost heave filled pores, no filler macroscopic frost heave occurred, and soil was insensitive to frost heave. For filler filling rates exceeds 0.25, as the filling rate increased, the filler frost heave rate increased significantly, and the soil

3. A microheave filler frost heave model, *ηmixed material* = *η*(*N*, *φ*, *C*, *α*, *β*0, *ds*, *ρw*, *w*,*ρmixed*, *θ*, *Gskeleton*), which considers the overlying load, filler plastic deformation friction angle, filler cohesion, filler frost heave rate, mixed material skeleton pore filling rate by the initial filler, mixed material ratio, water density, mixed material water content, mixed material density, filler content and mixed material skeleton ratio, was created. The applicability of the

This study was sponsored by the Foundation of China Academy of Railway

*β* mixed material skeleton grain pore filling rate by filler grains

microheave filler frost heave model was verified via testing.

effects on filler frost heave.

*DOI: http://dx.doi.org/10.5772/intechopen.82575*

*Frost Heave Deformation Analysis Model for Microheave Filler*

was sensitive to frost heave.

Sciences for the Youth (No.2017YJ050).

*α* filler expansion rate

*ds* mixed material ratio

*Gskeleton* skeleton grain ratio

*ηfiller* filler compactness *θ* filler content

*mfiller* quality of filler

*N* upper load *ρfiller* density of filler

*mmixed material* quality of mixed material *mskeleton* quality of skeleton

*ρmixed material* density of mixed material *ρskeleton* density of skeleton

*β*<sup>0</sup> initial filling rate by filler grains

Δ*Eelastic potential energy* D-value of elastic potential energy

Δ*h* overall specimen frost heave in mm

*η* microheave filler frost heave rate *ηmixed material* mixed material frost heave rate

*Hf* frozen depth (excluding frost heave) in mm

*K* soil internal friction coefficient, *K* = tan*φ M*<sup>1</sup> skeleton grain mass for a unit specimen volume

*M*<sup>2</sup> filler mass for a unit specimen volume

Δ*H* expansion potential energy

**Acknowledgements**

**Notes**

**117**

**Figure 15.** *Variations in frost heave rate with water content.*


**Table 5.** *Model parameters.*

the soil still produced a certain level of frost heave. The filler overlying load test showed that an overlying pressure produced a certain level of suppression effects on filler frost heave.

