Investigation on desiccation cracking behavior of clayey soils with a perspective of fracture mechanics: a review

Abstract

Purpose

Soil cracking is a common natural phenomenon. The existence of soil cracks has significant effects on the engineering properties of clayey soil, and can cause significant problems in geotechnical, geological, and environmental aspects. Understanding of the potential mechanisms of soil cracking is essential in assessments of potential damages to earthen infrastructures.

Materials and method

We review the past research efforts devoted to the experimental investigations and applications of fracture mechanics in soil cracking, attempting to provide a better understanding of the formation mechanism of desiccation cracking with a perspective of fracture mechanics.

Results and discussion

This review analyzes the influence of soil cracking on soil engineering properties and the significance of soil cracking phenomena. Past and current formulations of soil fracture criteria and their experimental investigation are discussed. This review reveals the factors that affect the mechanisms of soil fracture can be divided into two groups, namely soil intrinsic properties and test-related factors. The applications of fracture mechanics in soil cracking are also discussed with particular focus on soil fracture models that are separately based on linear elastic fracture mechanics (LEFM), elastic plastic fracture mechanics (EPFM), and numerical simulations of soil cracking based on fracture mechanics. Some challenges and prospects of the applications of fracture mechanics in soil cracking are presented.

Conclusions

Fracture mechanics is a significant method to explain soil crack initiation and propagation. It is expected that researchers can gain better understanding of the range of fracture mechanics applications in soil cracking, and seek improvements and extensions of existing models through this review.

1 Introduction

Some global regions have become more arid due to climate change. Desiccation cracking of soil is a common phenomenon in such regions. The formation of soil desiccation cracks due to moisture evaporation and volumetric shrinkage is frequently observed in drying climatic regions (Fig. 1). The existence of soil cracks has significant effects on the engineering properties of clayey soil, and can cause significant problems in geotechnical, geological and environmental aspects (Corte and Higashi 1960; Albrecht and Benson 2001; Sanchez et al. 2013; Tang et al. 2008; Tollenaar et al. 2017; Song and Cui 2020). Soil cracking enlarges soil macropore sizes, thereby, changing soil internal structure and hydraulic properties by creating preferential flow paths for the migration of moisture and contaminants (Albrecht and Benson 2001; Zhang et al. 2016; Abd El-Halim 2017; Tang et al. 2019; Wei et al. 2020; Xu et al. 2020; Colombi et al. 2021). The development of soil cracks can increase the hydraulic conductivity drastically and facilitate water infiltration, which will lead to failure of waste containment structures (Li and Zhang 2010, 2011). Water infiltration will change porewater pressure in unsaturated soils, thereby affecting the soil shear strength. In addition, cracks can form part of a slip surface that has no shear strength. (Li et al. 2011, 2017). Since clayey soils are widely used as impermeable layers for the landfill and waste isolation near the ground surface, soil cracks can greatly increase the hydraulic conductivity, which may cause the leakage of contamination and increase the risk of environmental pollution (Omidi et al. 1996; Miller et al. 1998; Hewitt and Philip 1999; Rayhani et al. 2007; Narani et al. 2020). Moreover, soil cracking increases clayey slope weathering depth and surface roughness. This aggravates soil erosion on the slope surfaces and also reduces soil cohesion, leading to slope instability (Baker 1981; Fredlund 1987; Alonso et al. 1995; Krisnanto et al. 2014; Wang et al. 2018a; Di Carlo et al. 2019; Poulsen et al. 2020; Stirling et al. 2020). Existence of soil cracks can degrade hydraulic-physical properties of earth structures (e.g., levee, slope and landfill), and can result in catastrophic failures as presented in Fig. 2. For example, there are thousands of kilometers of levees that have cracked in several countries, including the UK, the Netherlands, and the USA under seasonal drought conditions (Vahedifard et al. 2016). Billions of dollars have to be expended to repair cracked levees to prevent piping and direct levee failure every year (Robinson and Vahedifard 2016). Understanding of the potential mechanisms of soil cracking is essential in assessments of potential damages to infrastructures (Peron et al. 2009b; Cordero et al. 2017; Murray and Tarantino 2019; Wei et al. 2016, 2020).

Fig. 1

Drying-induced soil cracking in natural environment

Fig. 2

Potential undermining mechanisms imposed on the levee, slope and landfill due to drought-induced soil cracking

Currently, research methods used in analyses of soil cracking include laboratory tests, field tests and numerical simulations. In laboratory tests, soil cracking is controlled by a large number of factors, including temperature, thickness of soil layer, drying-wetting cycles, boundary effects and other experimental conditions (Tang et al. 2008, 2011a, 2013; Costa et al. 2013a; Xing et al. 2017; Lakshmikantha et al. 2018; Zeng et al. 2019). To study thermal effects on the soil desiccation cracking, Tang et al. (2008) performed desiccation experiments on a Nanjing area clayey soil. The soil cracking patterns obtained under different temperature conditions (30, 40, 50 ℃) were quantitatively indexed for comparison. The results showed that crack length, width, aggregate area increase with increasing temperature. Similar observations were made on other studied by Tang et al. (2011a). In order to analyze the influence of soil layer thickness on soil cracking, Zeng et al. (2019) analyzed clayey soil from Nanjing City. Saturated slurry samples were made with different soil layer thicknesses (0.5 cm, 1.0 cm, 1.5 cm). After drying, the crack total length, crack numbers, clod numbers and crack segment numbers were found to decrease with increase in soil thickness while the soil surface crack ratio and crack average width displayed on the opposite trend. Soil cracking during different wetting–drying cycles has been investigated in a number of studies (Wang et al. 2017; Albrecht 2001; Tang et al. 2011b). Wang et al. (2017) investigated the geometric and fractal features of soil cracking patterns using a digital camera on compacted soil samples subjected to four wetting–drying cycles. It was found that the crack networks of samples tend to be stable as the number of wetting–drying cycles increases and the crack patterns are mainly affected by the first cracking incident in the subsequent wetting–drying cycles. The same phenomenon has also been observed by other researchers (Albrecht 2001; Tang et al. 2011b; Wang et al. 2017, 2018b; Julina and Thyagaraj 2020). Moreover, effects of boundary conditions (i.e., sample sizes, shapes, and aspect ratios of specimens and containers) on the development of crack processes and the final crack patterns have been investigated (Lakshmikantha et al. 2018; Qu et al. 2020; Al-Jeznawi et al. 2021).

Due to large differences between experimental conditions of laboratory tests and the natural environment, the reliability and representativeness of the laboratory tests on soil cracking has often drawn controversy. Meanwhile, small-size laboratory tests may not represent well field conditions where the soil often contains coarse particles and structures, and has strong interactions with the atmosphere (Li and Zhang 2011a). Therefore, it is necessary to perform field experiments to verify natural soil crack patterns. However, the latter approach is hampered by high operation costs, time and effort inputs, and uncontrollable environmental conditions. For these reasons, few field studies on soil desiccation tests have been reported so far. Konrad and Ayad (1997a) excavated soil layers at three different levels and exposed them to continuous evaporation for 35 days. In order to assess cracking mechanisms, they observed and recorded the crack initiation and formation process of soil cracks in the topsoil layer, a weathered clay crust, and an intact clay. Abou Najm et al. (2010) conducted a 54-day field experiment on a farmland to monitor the evolution of surface cracks using digital imagery and 3-D visualization of the preferential flow paths network in soils using liquid latex. Li and Zhang (2010) investigated the field crack patterns and probability distributions of the geometric parameters of cracks and to determine the representative elementary volume (REV) of the crack network. Li and Zhang (2011) performed a 2-year field experiment to study the mechanisms of desiccation crack development in residual soils and backfill of road slopes. It was found that the crack pattern is closely related to the water content and drying time of the in-situ soil materials. Yu et al. (2021) conducted a desiccation crack monitoring campaign on a full-scale, vegetated infrastructure embankment subjected to 1-year of seasonally variable weather and pointed out the distribution of cracking was heavily related to prevailing wind direction (exposure) and crack response is driven by seasonal changes in meteorological and soil hydrological conditions.

As regards numerical simulation, Peron et al. (2009a) developed a two-dimensional Discrete Element Method (DEM) to study mud crack patterns under free boundary and constrained boundary conditions. The model was able to capture crack initiation, extension and final geometric morphology. Shin and Santamarina (2011) used a Finite Element Method (FEM) model to capture crack growth based on an effective-stress formulation, which follows only mode I tensile failure and is not able to account for crack interactions. Sima et al. (2014) proposed an approach to modeling of soil cracking in thin clay layers based on DEM, which could successfully capture the formation and development of cracks during drying. Hirobe and Oguni (2017) proposed a mathematical model to simulate patterns of soil cracks. This coupled model that they developed includes three aspects: desiccation, deformation, and fracture and is able to recast the basic features of the crack geometry. Lin et al. (2020) used DEM to simulate desiccation cracks initiated from both the top surface and the bottom in thin soil layers.

Although researchers have studied soil cracking processes and resulting crack patterns by different methods including laboratory tests, field tests and numerical simulations, no parameter can solely explain the soil cracking mechanism. Soil is a complicated, multi-phase granular material, and its cracking behavior is governed by several factors that defy the competence of any unified theoretical model in soil mechanics or fracture mechanics as regards soil crack initiation and propagation. In this review, the latest findings on the contribution of fracture mechanics to the analyses of soil cracking mechanism are presented in seven sections as follows: Section 1 expounds the significance of studying soil cracking and points out the lack of soil cracking mechanics models. Section 2 briefly introduces the fracture criterion of soil fracture parameters. Section 3 summarizes the past and existing experimental investigations of soil fracturing, including three-point bending test, four-point bending test, compact tension test, ring test, and compression test. Section 4 categorizes major factors that affect soil fracturing parameters into two groups, covering soil intrinsic properties and test-related factors. Section 5 discusses existing soil cracking models based on LEFM and EPFM. Section 6 describes numerical simulations of soil cracking based on fracture mechanics with primary focus on LEFM model and the cohesive crack method. Finally, a brief summary of concluding remarks and some prospects that require further investigations are presented in Sect. 7. Through this review, it is expected that researchers can better appreciate opportunities for adaptation of fracture mechanics principles for prediction of fracture patterns and development remedial measures for threatened natural and constructed systems.

2 Fracture criterion and soil fracture parameters

Over the past few decades, the field of fracture mechanics has undoubtedly prevented a substantial number of structural failures (Goehring et al. 2015). Fracture mechanics, known as the mechanics of cracked body, is the main tool for studies of cracking processes and behavior of materials. Heretofore, the development of fracture mechanics has focused on brittle and quasi-brittle materials such as concrete and rock (Cai et al. 1990; Scavia 1990; Han et al. 2018; Guan et al. 2019; Lau et al. 2019; Khalilpour et al. 2019), it has also been used to analyze the cracking of hardened soils. Skempton et al. (1969) and Bishop (1971) were the first researchers to utilize fracture mechanics to analyze the progressive failure of slopes composed of stiff but cracked clay, an approach that was followed up by Vallejo (1994). Konrad and Ayad (1997b) proposed a highly idealized analytical model based on LEFM, for predicting the spacings among the soil desiccation cracks and describing crack propagation. Numerous beam bending tests and compact tension tests have been performed on soil specimens to investigate the relationship between fracture toughness and tensile strength or other mechanical properties of soil (Wang et al. 2007a, 2007b; Lakshmikantha et al. 2008, 2012).

In fracture mechanics, cracks can be divided into single-mode cracks and mixed-mode cracks. The single mode fracture can be divided into three basic modes (Chaves et al. 2014; Wu et al. 2017): mode I (opening mode), mode II (shearing mode) and mode III (tearing mode), as illustrated in Fig. 3. In the Opening mode, the external load is the normal stress in the vertical crack plane. The relative displacement of the crack faces is perpendicular to the crack plane, which usually occurs when the load and geometry are symmetrical to the crack plane. In the Shearing mode, the external load is the in-plane shear stress parallel to the crack faces and perpendicular to the crack front. The crack slides perpendicular to the front edge of the crack in its own plane. In the Tearing mode, the external load is the out-of-plane shear stress parallel to the crack faces and parallel to the front edge of the crack. The crack displaces parallel to the front edge of the crack in its own plane. The mixed mode is a combination of two or the all three basic modes, and most of actual engineering cracks are mixed mode cracks.

Fig. 3

Three modes of cracking

In order to understand the soil cracking behavior based on fracture mechanics, it is important to presage the soil fracture parameters and corresponding fracture criteria. The primary soil fracture parameters are fracture toughness K_C, critical energy release rate G_C, J integral J_C, crack tip opening displacement CTOD, critical maximum hoop stress ${\sigma }_{\theta maxC}$ and critical strain energy density factor S_C. Fracture criteria that are based on the parameters listed above can be used to determine the initiation and development of soil cracks.

2.1 Fracture toughness K_C

Irwin (1957) proposed that crack propagation is related to the stress field near the crack tip and that the stress intensity factor can be used in setting the fracture criterion. This principle has been widely used in engineering practice till date. The stress field of the crack tip is illustrated in Fig. 4a. Essentially, K_I, K_II and K_III are the stress intensity factors of mode I, II ,III cracks, which can be expressed as Eq. (1):

Fig. 4

(a) Stress field of the crack tip, (b) the integration path of J integral

$$\begin{cases}{K}_{I}=\underset{r\to 0}{\mathrm{lim}}\sqrt{2\pi r}{\sigma }_{y}\left(r,0\right)\\ {K}_{II}=\underset{r\to 0}{\mathrm{lim}}\sqrt{2\pi r}{\tau }_{xy}\left(r,0\right)\\ {K}_{III}=\underset{r\to 0}{\mathrm{lim}}\sqrt{2\pi r}{\tau }_{yz}\left(r,0\right)\end{cases}$$

(1)

It is intuitive to find from Eq. (1) that the stress intensity factor K is related to the crack stress and length and increases with increase in stress. Soil cracks will rapidly propagate from the original state when the critical condition is exceeded. Commonly used methods for determining stress intensity factors are analytical methods, numerical calculation methods and experimental methods. The fracture criteria of modes I, II, III crack can be expressed as Eq. (2) based on stress intensity factors:

$$\begin{cases}{K}_{I}\ge {K}_{IC}\\ {K}_{II}\ge {K}_{IIC}\\ {K}_{III}\ge {K}_{IIIC}\end{cases}$$

(2)

where K_IC K_IIC, K_IIIC are the fracture toughness of mode I, II, III crack, respectively. Fracture toughness is a unique property of the material itself. It characterizes the resistance of the material to cracking.

2.2 Critical energy release rate G_C

The energy release rate G is an important parameter for analyzing soil cracking from an energy perspective, and refers to the energy released when a crack expands per unit area. The relationship between the energy release rate and the stress intensity factor can be obtained by analyzing the stress field at the crack tip from an energy perspective. It is can be expressed as Eq. (3):

$$\begin{cases}{G}_{I}=\frac{1-{\nu }^{2}}{E}{K}_{I}^{2}\\ {G}_{II}=\frac{1-{\nu }^{2}}{E}{K}_{II}^{2}\\ {G}_{III}=\frac{1+\nu }{E}{K}_{III}^{2}\end{cases}$$

(3)

where E is the elastic modulus, v is the Poisson's ratio. For the mixed-mode crack, the energy release rate is expressed as:

$$G=\frac{1-{\nu }^{2}}{E}{K}_{I}^{2}+\frac{1-{\nu }^{2}}{E}{K}_{II}^{2}+\frac{1+\nu }{E}{K}_{III}^{2}$$

(4)

The fracture criterion based on energy release rate can be expressed as Eq. (5):

$$G\ge {G}_{C}$$

(5)

2.3 J integral J_C

Cherepanov (1967) and Rice (1968) independently proposed an integral method that is independent of the integration path in elasto-plastic fracture mechanics, named J integral. It is used to comprehensively measure the stress–strain field strength at the crack tip. The J integral explains the change in potential energy due to crack propagation. This theory avoids the direct calculation of the elasto-plastic stress and strain fields near the crack tip and applies the far-field J integral as the average parameter of the stress–strain concentration characteristics of the crack tip. For the two-dimensional problems, J integral can be defined as:

$$J={\int }_{\Gamma }\left(Wdy-{T}_{i}\frac{\partial {u}_{i}}{\partial x}ds\right)$$

(6)

where $\Gamma$ is a simple integration path from a point on the lower surface of the crack to a point on the upper surface of the crack; W is the elastoplastic strain energy density; ${\text{T}}_{\text{i}}$ is the principal stress acting on the unit perimeter of the integration circuit; ${\text{u}}_{\text{i}}$ is the displacement on the boundary of the integral circuit; x is the cartesian coordinate; ds is the arc length of the integral circuit, as shown in Fig. 4b. It can be proved analytically that the J integral has nothing to do with the selection of the integral route $\Gamma$, that is, the J integral satisfies the path conservation. The J integral has a clear physical meaning, which represents the energy release rate of the crack when it grows along the x direction per unit length.

The fracture criterion based on J integral is expressed by Eq. (7):

$$J\ge {J}_{C}$$

(7)

where J_C is the strain fracture toughness of the crack during crack initiation. Per the principles of LEFM, it can be proven that the J integral, the energy release rate G, and the stress intensity factor K satisfy the following relationship:

$$J=G=\frac{{K}_{I}^{2}+{K}_{II}^{2}}{E}+\frac{1+\nu }{E}{K}_{III}^{2}$$

(8)

2.4 Crack tip opening displacement CTOD

Wells (1961) first proposed a fracture criterion that is based on CTOD and is expressed as a certain deformation dimension of the crack tip. The CTOD is a macroscopic parameter and is defined as the displacement between the crack faces of a loaded ideal crack. The fracture criterion based on crack tip opening displacement is expressed as Eq. (9):

$$\delta \ge {\delta }_{C}$$

(9)

where $\delta$ is the crack tip opening displacement; ${\delta }_{C}$ is the critical crack tip opening displacement, which is not only equivalent to the crack propagation resistance, but also a constant parameter of the material elastic–plastic fracture toughness. The specific definition of CTOD depends on which point of the crack faces is used for calculation. The representative definition methods mainly include the following:

Due to the material follows the linear elastic constitutive relation, the displacement field of the mode I crack tip can be obtained by elastic mechanics:

$$\left\{\begin{array}{c}u=\frac{{K}_{I}}{2G}\sqrt{\frac{r}{2\pi }}cos\frac{\theta }{2}\left(\kappa -1+2{\mathrm{sin}}^{2}\frac{\theta }{2}\right)\\ \upsilon =\frac{{K}_{I}}{2G}\sqrt{\frac{r}{2\pi }}sin\frac{\theta }{2}\left(\kappa +1-2{\mathrm{cos}}^{2}\frac{\theta }{2}\right)\end{array}\right.$$

(10)

where u and $\upsilon$ are the displacement components in the x and y directions; G is the shear modulus; the relationship between $\kappa$ and Poisson's ratio $\nu$ is expressed as Eq. (11):

$$\kappa=\left\{\begin{array}{cc}3-4v&\;Plaine\;strain\\\frac{3-v}{1+v}&\;Plane\;stress\end{array}\right.$$

(11)

while $\theta =+\pi \in$ on the crack faces:

$$u=2\upsilon =\frac{\kappa +1}{G}\frac{{K}_{I}\sqrt{r}}{\sqrt{2\pi }}$$

(12)

Irwin (1957) first gave a preliminary estimate of the plastic zone size at the mode I crack tip. The tensile stress in r* at crack tip just reaches the yield strength ${\sigma }_{ys}$ where r* is the size of the plastic zone, which can be calculated as:

$${r}^{*}=\frac{{K}_{I}^{2}}{2\pi {\sigma }_{ys}^{2}}$$

(13)

With the modification of the Irwin plastic zone, the true crack length is replaced by the effective crack length. Consequently, the origin moves to the tip of the effective crack, as shown in Fig. 5a. Therefore, the corrected CTOD value can be expressed as:

Fig. 5

(a) Definition of CTOD by Irwin method, (b) Dugdale model for the effective crack length

$$\delta =\frac{4{K}_{I}^{2}}{\pi E{\sigma }_{ys}}$$

(14)

Similarly, the effective crack length can also be obtained by the method that Dugdale (1960) proposed, as presented in Fig. 5b. The CTOD calculated by Dugdale method, is:

$$\delta =\frac{8{\sigma }_{ys}a}{\pi E}ln\left(\mathrm{sec}\left(\frac{\pi \sigma }{2{\sigma }_{ys}}\right)\right)$$

(15)

where a is the crack length, ($\sigma$) is the far-field tensile load.

The definition of CTOD and CTOA is the crack tip opening displacement and the crack tip opening angle, respectively. CTOD (or CTOA) has become a reliable and convenient fracture criterion to assess crack extension and directionality in materials. Many materials have been found to exhibit a constant critical CTOD (or CTOA) from crack initiation through failure, making CTOD (or CTOA) a useful and reliable fracture criterion (Newman et al. 2003; Zhu and Joyce 2012).

2.5 Critical maximum hoop stress ${\sigma }_{\theta maxC}$

The maximum hoop stress criterion is one of the representative composite fracture theories (Erdogan and Sih 1963). In the case of mode I and mode II mixed loading, the polar coordinate components of the crack tip singular stress field can be expressed as:

$$\begin{cases}{\sigma }_{r}=\frac{{K}_{I}}{2\sqrt{2\pi r}}\left(3-\mathrm{cos}\theta \right)cos\frac{\theta }{2}+\frac{{K}_{II}}{2\sqrt{2\pi r}}\left(3\mathrm{cos}\theta -1\right)sin\frac{\theta }{2}\\ {\sigma }_{\theta }=\frac{{K}_{I}}{2\sqrt{2\pi r}}\left(1+\mathrm{cos}\theta \right)cos\frac{\theta }{2}-\frac{3{K}_{II}}{2\sqrt{2\pi r}}sin\theta cos\frac{\theta }{2}\\ {\tau }_{\theta }=\frac{{K}_{I}}{2\sqrt{2\pi r}}sin\theta cos\frac{\theta }{2}+\frac{{K}_{II}}{2\sqrt{2\pi r}}\left(3\mathrm{cos}-1\right)cos\frac{\theta }{2}\end{cases}$$

(16)

where ${\sigma }_{r},{\sigma }_{\theta },{\tau }_{r}$ are the radial stress, hoop stress and shear stress near the crack tip, respectively; θ is the polar angle; r is the polar diameter. Therefore, the maximum hoop stress ${\sigma }_{\theta max}$ can be expressed as:

$${\sigma }_{\theta \mathrm{max}}=\frac{1}{2\sqrt{2\pi {r}_{0}}}\mathrm{cos}\frac{{\theta }_{0}}{2}\left({K}_{I}\mathrm{cos}\frac{{\theta }_{0}^{2}}{2}-3{K}_{II}\mathrm{sin}\frac{{\theta }_{0}}{2}\mathrm{cos}\frac{{\theta }_{0}}{2}\right)$$

(17)

where ${\theta }_{0}$ and ${r}_{0}$ are the critical values of polar angle and polar diameter, respectively. The fracture criterion based on maximum hoop stress can be expressed as Eq. (18):

$${\sigma }_{\theta \mathrm{max}}\ge {\sigma }_{\theta \mathrm{max}C}$$

(18)

2.6 Critical strain energy density factor ${S}_{C}$

For mixed-mode cracks, another representative fracture criterion, is the strain energy density factor criterion which was proposed by Sih (1974) based on the local strain energy density field. The strain energy density factor fracture criterion should satisfy not only the minimum strain energy density factor reaching the critical value, but also the initiation of a crack along the direction of the smallest strain energy density factor. The expression of strain energy density factor S is:

$$S={a}_{11}{K}_{I}^{2}+2{a}_{12}{K}_{I}{K}_{II}+{a}_{22}{K}_{II}^{2}+{a}_{33}{K}_{III}^{2}$$

(19)

where ${\text{a}}_{11}$, ${\text{a}}_{12}$, ${\text{a}}_{22}$, ${\text{a}}_{33}$ are intermediate parameters, which can be obtained by Eq. (20):

$$\begin{cases}{a}_{11}=\frac{1}{16\pi E}\left(\left(3-4v-\mathrm{cos}\theta \right)\left(1+\mathrm{cos}\theta \right)\right)\\ {a}_{12}=\frac{1}{8\pi E}sin\theta \left(\mathrm{cos}\theta -\left(1-2v\right)\right)\\ {a}_{22}=\frac{1}{16\pi E}\left(4\left(1-v\right)\left(1-\mathrm{cos}\theta \right)+\left(1+\mathrm{cos}\theta \right)\left(3\mathrm{cos}\theta -1\right)\right)\\ {a}_{33}=\frac{1}{4\pi E}\end{cases}$$

(20)

Sih (1974) suggested that the crack propagation direction was along the minimum value of the strain energy density factor. The fracture criterion based on strain energy density factor can be expressed as Eq. (21):

$$S\ge {S}_{C}$$

(21)

3 Experimental investigation of soil fracturing

Section 2 focuses on fracture criterion and the significant of soil fracture mechanics parameters. In order to obtain these fracture mechanics parameters, researchers have performed numerous experimental investigations on soil fracturing. Among them are three-point bending tests, four-point bending tests, compact tension tests, ring tests, compression tests and other methods. Typical experimental methods for investigating soil fracturing behavior are summarized in Table 1.

Table 1 Summary of experimental studies on fracture parameters of various soil materials

3.1 Three-point bending test

Following ASTM International (ASTM E399-20 2020), the standard proportions and tolerances of the single-edge notched beam (SENB) is shown in Fig. 6, which can be used as the standard specimen to determine the plane strain fracture toughness of the material including soil, metal, concrete, ceramics, and rock (Wang et al. 2007a). However, this standard is more suitable for brittle materials, but the soil is not a completely brittle material. For a given state of a particular soil, there is no specific standard method in measuring soil fracture toughness up to now. It is recommended to measure the soil fracture toughness according to the method provided by the ASTM International (ASTM E399-20 2020). In order to ensure the linear elastic behavior and minimize the size effect of the specimen, the following criteria are used on the sample size and initial crack length:

$$\begin{cases}\left(W-a\right)\ge 2.5{\left(\frac{{K}_{I}}{{\sigma }_{ys}}\right)}^{2}\\ a\ge 2.5{\left(\frac{{K}_{I}}{{\sigma }_{ys}}\right)}^{2}\\ B\ge 2.5{\left(\frac{{K}_{I}}{{\sigma }_{ys}}\right)}^{2}\end{cases}$$

(22)

where W is the width of specimen, B is the thickness of specimen. Sture et al. (1999) proposed that when the a/W of cemented sand is in the range of 0.2–0.55, the stress intensity factor K of the three-point bending specimen can be calculated according to the following Eq. (23):

$${K}_{I}=\frac{PS}{B{W}^{3/2}}{f}_{1}\left(\frac{a}{W}\right)$$

(23)

where S is the effective length of specimen, P is the concentrated load applied on the specimen, f₁ is a function of a/W, which can be express as:

Fig. 6

Standard single edge cracked beam (ASTM E399-20, 2020)

$${f}_{1}\left(\frac{a}{W}\right)=\frac{3{\left(\frac{a}{W}\right)}^{1/2}\left(1.99-\left(\frac{a}{W}\right)\left(1-\frac{a}{W}\right)\left(2.15-3.93\frac{a}{W}+2.7\frac{{a}^{2}}{{W}^{2}}\right)\right)}{2\left(1+\frac{2a}{W}\right){\left(1-\frac{a}{W}\right)}^{3/2}}$$

(24)

A number of researchers have used the standard three-point bending test to successfully determine the fracture toughness or crack tip opening displacement (CTOD) of soil specimens (Li and Yang 2000; Aluko and Chandler 2006; Zhang et al. 2008; Lenci et al. 2012). For example, Lenci et al. (2012) investigated the rupture behavior of the unfired dry earth by the testing apparatus shown in Fig. 7a. The three-point bending test was carried out under monotonic and cyclic loading and the related force–displacement diagrams were theoretically interpreted to characterize the fracture parameters (K_IC and CTOD) of the soil material.

Fig. 7

Four geometrical configurations of the three-point bending test (Hallett and Newson 2001; Wang et al. 2007a, 2020; Lenci et al. 2012)

In the standard three-point bending test, the weight of samples will have an effect on the accuracy of testing results, even if the specimen may not rupture failure under the action of its weight. Compared with the concentrated load, the self-weight of the specimen is not an insignificant quantity that can be ignored. Furthermore, the bending moment generated by the specimen self-weight at the crack face, changes continuously with the deformation of the specimen during the bending test.

In order to minimize or remove the impact of the specimen self-weight on the test results, researchers have utilized a series of improved three-point bending test apparatuses in their studies. Two retractable supports were installed by Chandler (1984) on both sides of the initial crack at the specimen bottom, which can symmetrically balance the specimen self-weight. The results show that the tested soil behaves like a tough metal as the internal crack growth resistance increases remarkable with crack growth. As illustrated in Fig. 7b, Hallett and Newson (2001) installed two glass slides on the bottom of the specimen that can rotate around the rollers. Weights were placed on both ends of the glass slides to balance against the soil specimen self-weight. The same balancing principle is used in Yoshida and Hallett (2008) and Amarasiri et al. (2011). Wang et al. (2007a, 2007b, 2016) rotated the bottom side of the original specimen by 90° to make the direction of the load P perpendicular to the weight direction of the soil specimen (shown in Fig. 7c), which not only prevented the soil specimen from fracturing solely under its self-weight, but also makes the variation curve easy to obtain after the peak load. Actually, the influence of the specimen self-weight cannot be completely eliminated, because the specimen is a soil beam with a specific thickness.

In order to minimize the disturbance that crack prefabrication and sample transfer can impose on a sample due to the low strength of the soil during test preparation, Wang et al. (2020) proposed the use of notched semi-circular bend specimens (NSCB) for three-point bending tests on account of their small dimensions and ease of testing as shown in Fig. 7d. According to LEFM, the initial crack starts to fracture from the crack tip under the action of concentrated load P. The K_IC value of the NSCB specimen can be determined by the following Eqs. (25) and (26):

$$K_{IC}=Y\frac{P_{max}\sqrt{\pi a}}{2BR}(a/R\geq0.2)$$

(25)

$$Y=-1.297+9.516\frac{S}{2R}-\left(0.47+16.457\frac{S}{2R}\right)\frac{a}{R}+\left(1.071+34.401\frac{S}{2R}\right){\left(\frac{a}{R}\right)}^{2}$$

(26)

where P_max is the critical value of concentrated load, Y is the nondimensionalize stress intensity factor (SIF) corresponding to specimen geometry, R is the radius of semicircular specimen.

3.2 Four-point bending test

The three-point bending test is mainly used to determine fracture toughness in mode I cracking, but the actual cracking mode of the soil is usually more complicated. It could be mode I, mode II or mixed mode. Therefore, it is not enough to solely study mode I cracking of soil. In addition to the three-point bending test, the four-point bending test is an important method for determining the fracture toughness in mode II cracking and the stress intensity factors of mixed mode I-II cracking during the fracture propagation of soil specimens. Wang et al. (2007a) proposed an improved four-point asymmetric bending test configuration with horizontal concentrated load, as presented in Fig. 8a. The shear force and bending moment that act on the crack plane, can be obtained based on the force and moment equilibrium theories drawn from in engineering mechanics, which can be expressed as Eq. (27):

$$\begin{cases}Q=\frac{{L}_{1}-{L}_{2}}{{L}_{1}+{L}_{2}}P\\ M=\left({L}_{2}+c\right)\frac{{L}_{1}}{{L}_{1}+{L}_{2}}P-\left({L}_{1}+c\right)\frac{{L}_{2}}{{L}_{1}+{L}_{2}}P\end{cases}$$

(27)

where Q and M are the shearing and force bending moment on crack plane; P is concentrated load applied on the specimen; L₁ and L₂ the horizontal distance from the application point of concentrated load P to fulcrums A (or D) and B (or C), respectively; c is the horizontal distance from the application point of concentrated load P to the initiate crack. While the crack length a is 0.25 ~ 0.75 times as long as the specimen width W, the stress intensity factor K of the four-point bending specimen is calculated according to the following equation (Lin and Xue 1985):

$$\left\{\begin{array}{c}K_I=\frac M{BW^{3/2}}f_2\left(\frac aW\right)\\K_{II}=\frac Q{BW^{3/2}}f_3\left(\frac aW\right)\end{array}\right.\quad(a=0.25\sim0.75W)$$

(28)

where f₂ and f₃ are functions of a/W, which can be expressed as:

Fig. 8

Two geometrical configurations of four-point bending test and the stages of crack evolution in wet soil during the flexure test. (Hallett and Newson 2005; Wang et al. 2007a)

$$\begin{cases}{f}_{2}\left(\frac{a}{W}\right)=\frac{6{\left(\frac{a}{W}\right)}^{1/2}\left(1.99-\left(\frac{a}{W}\right)\left(1-\frac{a}{W}\right)\left(2.15-3.93\frac{a}{W}+2.7\frac{{a}^{2}}{{W}^{2}}\right)\right)}{\left(1+\frac{2a}{W}\right){\left(1-\frac{a}{W}\right)}^{3/2}}\\ {f}_{3}\left(\frac{a}{W}\right)=\left(1.442-5.08{\left(\frac{a}{W}-0.507\right)}^{2}\right)sec\frac{\pi a}{2W}\sqrt{\mathrm{sin}\frac{\pi a}{2W}}\end{cases}$$

(29)

If c is equal to zero, according to Eq. (27), the bending moment acting on the crack plane is also equal to zero. This means that the type of crack is a mode II crack. In this condition, the stress intensity factor for the mode II crack is obtained by using Eq. (28). When the concentrated load P is the critical load for crack instability propagation, the stress intensity factor is the K_IIC of the soil specimen. Li et al. (2000) used the same four-point bending device to measure the K_IIC of the soil. The behavior of the interface between the soil and concrete, K_IIC can be expressed as:

$${K}_{IIC}=\frac{{P}_{max}}{3B{W}^{1/2}}\left(1.442-5.08{\left(\frac{a}{W}-0.567\right)}^{2}\right)\mathrm{sec}\frac{\pi a}{2W}\sqrt{\mathrm{sin}\frac{\pi a}{2W}}$$

(30)

If c is unequal to zero, on the basis of Eq. (27), the shearing force and bending moment acting on the crack plane are also unequal to zero, which means that the type of crack is mixed mode I-II crack. When the concentrated load P is the critical load for fracturing failure, the stress intensity factors are the K_IC and K_IIC of the soil specimen.

Hallett and Newson (2005) collected data on load transmission, specimen bending, crack propagation and crack-mouth opening to evaluate the crack tip opening angle (CTOA) using a modified four-point bending test device (shown in Fig. 8b). In the experiment, a new method that is based EPFM was used to describe the formation of cracks in a plastic soil by the CTOA. As illustrated in Fig. 8c, crack evolution in wet soil during the flexure test can be divided into three stages. In the first stage (crack opening), once the yield point is exceeded, the crack will open such that the plastic strain energy will accumulate at the crack tip. During the second stage (crack initiation), the crack begins to propagate and the plastic strain energy be released with corresponding drop in the force applied. In the third stage (stable ductile crack growth), once the ductile crack propagation becomes stable, steady-state conditions will occur.

3.3 Compact tension test

Following ASTM International (ASTM E399-20 2020), the compact tension (CT) specimen (the standard configurations are shown in Fig. 9) can be used as another standard specimen to determine the plane strain fracture toughness of the material. This is composed of a single edge notched and fatigue pre-cracked plate under tensile load. The requirements concerning sample size and initial crack length have been mentioned in Eq. (22). The stress intensity factor K of the compact tension specimen is calculated based on the following Eq. (31):

$$K=\frac{P}{B{W}^{1/2}}{f}_{4}\left(\frac{a}{W}\right)$$

(31)

where f₄ is functions of a/W, which can be express as:

Fig. 9

Compact tension specimen with standard proportions (ASTM E399-20 2020)

$${f}_{4}\left(\frac{a}{W}\right)=\frac{\left(2+\frac{a}{W}\right)\left(0.886+4.64\frac{a}{W}-13.32{\left(\frac{a}{W}\right)}^{2}+14.72{\left(\frac{a}{W}\right)}^{3}-5.6{\left(\frac{a}{W}\right)}^{4}\right)}{{\left(1-\frac{a}{W}\right)}^{3/2}}$$

(32)

Konrad and Cummings (2001) used the fracture mechanics test procedure recommended by the ASTM Standard E399-83 to determine the fracture toughness of sand layer-base and crushed stone layer-base. The value of K_IC can be calculated by using Eq. (33) (ASTM E399-83, 1983):

$${K}_{IC}=\frac{{P}_{max}}{B{a}^{1/2}}{f}_{5}\left(\frac{a}{W}\right)$$

(33)

where f₅ is functions of a/W, which can be expressed as:

$${f}_{5}\left(\frac{a}{W}\right)=30.96\left(\frac{a}{W}\right)-195.8{\left(\frac{a}{W}\right)}^{2}+730.6{\left(\frac{a}{W}\right)}^{3}-1186.3{\left(\frac{a}{W}\right)}^{4}+754.6{\left(\frac{a}{W}\right)}^{5}$$

(34)

In order to study the relationship between tensile strength and fracture toughness of soil and the size effect of soil cracking, Lakshmikantha et al. (2008, 2012) used a direct tensile strength equipment and a CT-test equipment to obtain the two parameters. The schematic diagram of CT-test equipment is shown in Fig. 10. For convenience of comparison, two methods were used for processing fracture toughness data obtainable through Eq. (31) and Eq. (33). Prat et al. (2008) used the same method to determine the fracture toughness and tensile strength of soil specimens with different water contents, and formulated a numerical model to study the initiation and propagation of soil desiccation cracks based on soil mechanics and fracture mechanics.

Fig. 10

Schematic diagram of CT-test equipment (Lakshmikantha et al. 2008)

In addition to fracture toughness and stress intensity factors, the critical energy release rate can also be obtained using the compact tension (CT) test (Lee et al. 1988; Ayad et al. 1997). Ayad et al. (1997) suggested that the critical energy release rate due to crack growth corresponds to the difference in value between the rate of work done by external loading and the rate of increase in strain energy of the specimen. The definition of specimen compliance ($\lambda$) is as follows:

$$\lambda \text{=}\frac{{u}_{c}}{{F}_{c}}$$

(35)

where $F_{c}$  is the fracture load, $u_{c}$ is the load point displacement. The critical energy release rate can be expressed as:

$${G}_{C}=\frac{1}{2B}{F}_{c}^{2}\frac{d\lambda }{da}$$

(36)

3.4 Ring test

Due to plastic behavior near the crack tip, LEFM has limitations in analyzing the cracking of some non-brittle materials, especially, the desiccation crack of wet or slurried soils (Amarasiri and Kodikara 2011; Costa and Kodikara 2012). The size of the crack must be much larger than the size of the plastic zone, so that the crack body can be seen as a whole with approximately elastic manner (Janssen et al. 2004). Cracking problems that do not meet these required extreme conditions have to be studied using EPFM theory. Compared with LEFM, EPFM is a relatively complicated method of analyzing soil cracking. It has not been well developed so far. As discussed in Subsections 2.3 and 2.4, the J integral and CTOD are also two significant EPFM fracture parameters in the study of soil cracking. When the soil transforms from a highly plastic state to an elastic or quasi-brittle state due to desiccation, the J integral become a general fracture parameter for the soil. The ring test is mainly used to obtain the value of J integral, which has been widely used to evaluate the stress development, the strength changes and the cracking behavior of fiber reinforced concrete or polymer-modified asphalt concrete (Mindess et al. 1977; Velazco et al. 1980; Bhurke et al. 1997; Kuai et al. 2009).

Harison et al. (1994) used the ring test to determine the fracture toughness and the critical J integral of 132 compacted samples of two fine grained soils and compared the fracture toughness values measured by the bending test and the ring test. The configuration of ring test and crack-tip model are shown in Fig. 11a. The expression of J integral form is transformed into the integral area of the load and load point displacement curve (Rice et al. 1973):

Fig. 11

Two configuration of ring test (Harison et al. 1994; Costa and Kodikara 2012)

$$J=\frac{1}{B}{\int }_{0}^{\delta }\left(-\frac{\partial P}{\partial a}\right)d\delta$$

(37)

Rice et al. (1973) suggested that the total load point displacement of the cracked specimen should be separated into two parts, namely, the measurable displacement without the initial crack ${\delta }_{nocrack}$ and the displacement caused by the introduction of initial crack ${\delta }_{crack}$. If the integrated area of the P-load point displacement curve measured when the sample is free of cracks is A_nc, and the integrated area measured after the cracks are introduced is A_c, the J integral described by Eq. (37) can be written as following:

$$\begin{array}{c}J=\frac{{f}^{^{\prime}}\left(a\right)}{f\left(a\right)}\frac{1}{B}{\int }_{0}^{{\delta }_{crack}}Pd{\delta }_{crack}\\ =\frac{{f}^{^{\prime}}\left(a\right)}{f\left(a\right)}\frac{1}{B}\left({A}_{c}-{A}_{\text{nc}}\right)\end{array}$$

(38)

where f(a) is the function determined by the ratio of ${\delta }_{crack}$ to the load P, f′(a) is the derivative of f(a).

Costa and Kodikara (2012) proposed an innovative ring test to determine fracture properties (J integral) during the drying process of a clayey soil. The schematic diagram of slurry specimen is shown in Fig. 11b. This is the first time that fracture mechanics is applied to study the drying of slurry specimens. As presented in Fig. 11b, the J integral can be expressed as the sum of several line integrals based on the basic principles of fracture mechanics (Luo et al. 2003), is defined as:

$$J={J}_{V1}+{J}_{H2}+{J}_{V2}+{J}_{H1}$$

(39)

where

$$\begin{cases}{c}{J}_{V1}={\int }_{{y}_{1}}^{{y}_{2}}Wdy-\left({\sigma }_{xx}\frac{\partial {u}_{x}}{\partial x}+{\sigma }_{xy}\frac{\partial {u}_{y}}{\partial x}\right)dy\\ {J}_{V2}={\int }_{{y}_{2}}^{{y}_{1}}Wdy-\left({\sigma }_{xx}\frac{\partial {u}_{x}}{\partial x}+{\sigma }_{xy}\frac{\partial {u}_{y}}{\partial x}\right)dy\\ {J}_{H1}=-{\int }_{{x}_{2}}^{{x}_{1}}\left({\sigma }_{yy}\frac{\partial {u}_{x}}{\partial x}+{\sigma }_{yx}\frac{\partial {u}_{x}}{\partial x}\right)dx\\ {J}_{H2}=-{\int }_{{x}_{1}}^{{x}_{2}}\left({\sigma }_{yy}\frac{\partial {u}_{y}}{\partial x}+{\sigma }_{yx}\frac{\partial {u}_{x}}{\partial x}\right)dx\end{cases}$$

(40)

where V₁, V₂, H₁ and H₂ are the four paths of four line integrals, respectively; ${u}_{x},{u}_{y},\;and\;{\sigma }_{yy}$ are the displacements and stress in x and y directions, respectively.

It is worth mentioning that particle image velocimetry (PIV) analysis is used to obtain the strain and stress of the slurry specimen surface which can be later used to calculate the J integral. Then, according to the same method, Costa et al. (2013b, 2016) later not only calculated the J integral of Churchill clay and kaolin clay by using the new ring test, but also got the J integral of mud specimens and compacted soil specimens with the corresponding linear shrinkage coefficient. In addition, the independence of the J integral calculation path and the relationship between the J integral and the water content were also analyzed.

3.5 Other methods

In addition to the four test methods discussed above, there are some indirect methods exemplified by the compression test that can be used to determine the soil fracture parameters (Hallett et al. 1995; Nishimura and Shimizu 2004; Liu and Liu 2011). Centre cracked cylinder specimens with initial tension cracks were set in advance and made of three mixtures of silica sand and kaolinite. They were tested to determine fracture toughness by an indirect tension test (Hallett et al. 1995). The configuration of indirect tension test is shown in Fig. 12a. Hallett et al. (1995) found that the plastic zone (as presented in Fig. 12a) at the pre-existing crack tip was significant due to the crack propagation. This could explain the additional energy dissipation during crack development. In general, the relationship between the crack direction and the stress affects the stress intensity and failure mode of the crack tip. In the test, the failure mode of the pre-existing crack is of mode I type. Therefore, the fracture toughness is evaluated as follows:

Fig. 12

Two configurations of compression test (Hallett et al. 1995; Liu and Liu 2011)

$${K}_{IC}\text{=}Y{\sigma }_{f}{a}^{1/2}$$

(41)

where

$$\begin{cases}{\sigma }_{f}=\frac{{F}_{c}}{\pi {R}_{0}l}\\ Y\left(\frac{a}{2{R}_{0}}\right)=1+\frac{3}{2}{\left(\frac{a}{{R}_{0}}\right)}^{2}+\frac{3}{4}{\left(\frac{a}{{R}_{0}}\right)}^{6}+\frac{3}{64}{\left(\frac{a}{{R}_{0}}\right)}^{8}\end{cases}$$

(42)

Where ${\sigma }_{f}$ is the failure stress of the disc, R₀ is radius of the disc, l is length of the disc, Y is the specimen geometry stress intensity factor.

Liu and Liu (2011) used soil specimens containing wing cracks with different inclination angles to perform uniaxial unconfined compression fracture experiments. The specimen dimension and the location of external load are illustrated in Fig. 12b. The model of compression and shear fracture of wing cracks is also presented in Fig. 12b. It is noteworthy that the failure mode of the compression specimen is the mixed mode I-II cracking type. Therefore, the fracture toughness of the wing crack specimens including two failure modes (mode I and mode II), can be expressed as:

$$\left\{\begin{array}{c}{K}_{IC}=\frac{1}{2}{\sigma }_{f}\left(1+\mathrm{cos}2\beta \right)\sqrt{\pi a}\\ ={\sigma }_{f}\sqrt{\pi a}{\mathrm{sin}}^{2}\beta \\ {K}_{IIC}=\left(-\frac{1}{2}{\sigma }_{f}\mathrm{sin}2\beta \right)\sqrt{\pi a}\\ ={\sigma }_{f}\sqrt{\pi a}sin\beta cos\beta \end{array}\right.$$

(43)

where $\beta$ is the angle between the pre-existing crack and the failure stress ${\sigma }_{f}$ which also called the crack angle.

This section introduces the five main test methods used to determine soil fracture parameters. It is obvious that each method has its advantages and limitations. Although the three-point and the four-point bending tests are very convenient, the self-weight of the specimen in both tests affects the accuracy of the fracture parameters. Although some researchers have improved the test device based on different principles, they still cannot completely overcome this problem. Compact tension test is also one of the important methods to measure soil fracture parameters. Although it can remove the influence of its self-weight on the test results, it is likely that failure of the specimen loading point will precede the comprehensive failure of the specimen because of the low tensile strength of the soil. The ring test is mainly used to obtain the value of J integral based on EPFM, which is the first time that fracture mechanics is applied to the study of the drying process of slurry specimens. This test method has been widely applied to evaluate stress development. However, the procedures of ring test are too cumbersome for frequent use in the testing of soil cracking. Consequently, the ring test is rarely used. The compression test can also be used to measure soil fracture toughness, it is mainly used to test soils of the relatively high strength regime. Pre-existing cracks and external loading pattens of the compression specimens affect the accuracy of fracture parameters of soil samples, especially when the samples are of lower strength.

4 Factors that affect soil fracturing parameters

The fracture process of soil is complicated, because soils are composites of various grains of diverse minerals that interact at different levels of bond strength. External factors, including test protocols affect soil fracture as setting. Thus, the factors influencing soil fracture parameters can be categorized into two major groups, including soil intrinsic properties and test-related factors.

4.1 Soil intrinsic properties

Soil is a multi-phase material that consists of soil particles (solid phase), water (liquid phase) and air (gaseous phase). The existence of different phases leads to the inhomogeneity, a characteristic which also reflects other properties, including water content (w), dry density (p), clay content and so forth. Table 2 is a summary of research that has been performed on the effects of soil intrinsic properties on fracturing parameters.

Table 2 Influence of soil intrinsic properties on fracture parameters

4.1.1 Water content

Soil characteristics, especially for fine grained soils change with water content (Haynes and Swift 1990). Water does not exist in soil merely as another physical component but participates in complex physical–chemical interactions with soil particles. These interactions are partial determinants of the engineering properties of the soil. Changes in soil water content inevitably influence the ability of soil to resist fracturing and this is reflected in the value of soil fracture toughness (Wang et al. 2007a).

Figure 13 shows how water content influences the fracture toughness of soils. In addition to the results obtained by Wang et al. (2007a), other test results indicate that the value of fracture toughness decreases with increase in water content. As the compacted specimen is in an unsaturated state, the change of water content may result in the change of bond water thickness among soil particles. Meanwhile, the interaction force among soil particles and suction potential in the soil will have been changed (Amarasiri et al. 2011; Lakshmikantha et al. 2008, 2012; Wang et al. 2007a, 2007b, 2020).

Fig. 13

The dependence of fracture toughness on water content

4.1.2 Dry density

Dry density of a soil can be defined as the ratio of mass of solids to the total volume of a soil. This parameter is thus determined by the proportions of the solid phase soil particles and pores in the soil. When the relative density of soil particles is fixed, soil dry density reflects the total amount of pores in the soil, itself a reflection of the degree of compaction of the soil. Therefore, a change in dry density is usually a change in soil strength. These same factors affect fracture behavior of soil. Considering that most threats of soil fracture in geotechnical works involve compacted soils, the dry density is thus an important parameter in soil fracture (Wang et al. 2007b; Demagistri et al. 2018).

Figure 14 illustrates the dependence of fracture toughness on dry density. Wang et al. (2007a, 2007b, 2020) concluded that the value of fracture toughness increases linearly as specimen dry density increases. This is consistent with the findings on the desiccation cracking of compacted soil with different dry densities (Albrecht and Benson 2001). Volumetric shrinkage strain decreases with increasing compaction effort. Samples with the largest volumetric shrinkage strains typically contained the largest number of cracks. For the compacted soil with a certain water content, the bulk density and clay content increase with the increase of dry density. At the same time, as the dry density increases, the distance between the soil particles gradually decreases and the contact areas among soil particles increase, resulting in an increase in friction among the soil particles. Therefore, the higher the dry density of the specimen is, the more difficult the initial crack propagation will be.

Fig. 14

The dependence of fracture toughness on dry density

In addition to water content and dry density, some researchers have also studied the influence of clay, salinity and admixture content on soil fracture parameters (Hallett and Newson 2001, 2005; Zhang et al. 2008). The greater the proportion of sand in a soil mix, the smaller the crack tip opening angle (CTOA) will be as a reflection of the higher energy requirement for crack extension. To be specific, the specimens with ratios of sand to kaolinite of 75:25 and 50:50 were tested for bending. CTOA values of 0.19 mm⁻¹ to 0.24 mm⁻¹ were obtained, which indicate that more strain is needed for crack propagation (Hallett and Newson 2001). Furthermore, changing the ratio of silica sand to kaolinitic clay slightly from 0:100 to 20:80 reduces the CTOA by about 50%, from 0.230 mm⁻¹ to 0.136 mm⁻¹ (Hallett and Newson 2005).

Hallett and Newson. (2005) added a salt (0.5 M NaCl) to pure kaolinite in their tests and found that CTOA changed from 0.23 mm⁻¹ to 0.17 mm⁻¹. This means that the soil saline samples require more energy for crack development when compared with non-saline samples. Increase in salinity produces a reduction in soil surface crack ratio (Zhang et al. 2019). In other words, the distribution density of the crack networks on the soil surface is reduced with increased salinity, but more centralized (DeCarlo and Shokri 2014; Shokri et al. 2015; Zhang et al. 2016, 2019).

In order to quantify the effect of root exudates (polygalacturonic acid (PGA)) on the interparticle bond energy and fracture toughness of clay, Zhang et al. (2008) used pure kaolinite mixed with different PGA content to form test samples. Half of the test samples were washed repeatedly to remove unbound PGA and evaluate the persistence of its effects. The results show that soil fracture toughness increases exponentially with added PGA and that soil washing increases this trend. This use of fracture mechanics in analysis enables assessment of the stability of the rhizosphere and has implications in soil erosion and agriculture.

4.2 Test-related factors

The determination of soil fracture parameters is strongly dependent on test sample preparations and test conditions. Among these parameters are the ratio of initial crack length to specimen width (a/W), environmental temperature (T), loading rate (P_L), specimen thickness and so on. Many investigations have focused on the influence of these tested-related factors on the soil fracture behavior. Table 3 is a summary of how the test-related factors influence the soil fracture parameters.

Table 3 Influence of the test-related factors on soil fracture parameters

4.2.1 The ratio of initial crack length to specimen width

Fracture toughness is the resistance of soil to fracture. It is assigned to the state of the material, and has nothing to do with the change of a/W. Therefore, when the value of fracture toughness changes with the initial crack length, it should not be considered to truly reflect the ability of the soil to resist fracture failure.

As discussed in Sect. 3, the fracture toughness of soil sample is mainly determined by the bending beam test and the compact tension test. The fracture toughness obtained by the two methods for soil samples with different a/W values have been extensively investigated by researchers (Ayad et al. 1997; Konrad and Cummings 2001; Nishimura and Shimizu 2004; Lakshmikantha et al. 2008, 2012; Wang et al. 2016, 2020). The results show that the fracture toughness of soil roughly decreases with increase of a/W, but there is a stable stage on that trend when bending beam test is used (Fig. 15). As mentioned above, fracture toughness should be a fixed value for a given state of a particular soil. Therefore, a/W of the test sample should be limited in order to obtain the real value of fracture toughness (Nishimura and Shimizu 2004; Wang et al. 2016, 2020).

Fig. 15

The dependence of fracture toughness on the ratio of initial crack length to specimen width

4.2.2 Temperature

As for the influence of temperature on the fracture strength of soil, there are mainly researches on frozen soil. Frozen soil is composed of soil particles, ice inclusions, water (unfrozen water and tightly bound water) and gaseous inclusions (water vapor and air) (Lai et al. 2009, 2010). The difference between frozen soil and unfrozen soil is the existence of ice in the former (Chang and Liu 2013). The fracture parameters of frozen soil are strongly sensitive to ambient temperature. Many investigations have been performed on the fracture of frozen soils. (Li et al. 2000; Li and Yang 2000; Konrad and Cummings 2001; Liu and Liu, 2011). Generally, frozen soils are brittle at lower temperatures, and elastic–plastic at higher temperatures. As presented in Fig. 16, the values of K_IC and K_IIC increase as temperatures drop. This means that the strength of frozen soils increases as temperature decreases. Lowering of ambient temperature increases the strength and fracture toughness of a frozen soil. The reason is that the existence of ice will greatly increase the cohesion among particles.

Fig. 16

The dependence of fracture toughness on temperature

4.2.3 Loading rate

In geotechnical engineering practice, rarely are external loads applied instantaneously but tend to increase. For this reason, some researchers have studied the influence of loading rate on the soil fracture patterns (Li et al. 2000; Li and Yang, 2000; Liu and Liu, 2011). As shown in Fig. 17a–b, the values of both K_IC and K_IIC grow as the loading rate increases in the early stage. With increase loading time, fracture toughness becomes less sensitive to increase loading rate.

Fig. 17

The dependence of fracture toughness on loading rate. (a) stress controlled tests; (b) strain controlled tests

Apart from the test-related factors discussed above, some researchers have also conducted a series of investigations on other significant factors, including soil sample thickness and notch crack angle (Liu and Liu 2011; Wang et al. 2020). Wang et al. (2020) used semi-circular samples with different thicknesses to perform three-point bending tests and found that the value of fracture toughness slightly decreases with increase in thickness and stabilizes after the thickness is greater than 50 mm. The small differences observed justify the consideration of fracture toughness as being a material characteristic. Thus, the method proposed by Wang et al. (2007a) to remove the influence of specimen self-weight on the experimental results is practicable.

During compression process, the soil undergoes bending failure and compression failure. Under certain conditions, the influence of compressive fracture on the soil is dominant. In order to study the mechanical characteristics of the compressive fracture of the soil, Liu and Liu (2011) used compact compression samples with different tilted wing crack angles to perform uniaxial unconfined compression fracture tests (shown in Fig. 12b). As the tilted wing crack angle increases, the value of K_IC and K_IIC show two different growth trends. The values of K_IC and K_IIC increase linearly and logarithmically with the increase in crack angle, respectively.

5 Soil cracking model based on fracture mechanics

The existence of microcracks and grains within a soil leads to stress concentrations, which can result in the failure of the soil. In a soil sample, once the surface energy of the expanding crack is balanced with the elastic potential energy released around the crack, the initiation and propagation of the crack will occur (Griffith et al. 1924). Fracture mechanics was originally applied to study the problem of tensile cracking of soils. Defects in the interior of heterogeneous soils lead to decreased strength, and the stress concentrations generated at tips of microcracks cause them to develop into visible macrocracks (Hallett et al. 1995; Lima and Grismer 1994; Hallett and Newson, 2001; Prat et al. 2008).

5.1 Linear elastic fracture mechanics LEFM

LEFM is often used to construct the soil desiccation cracking prediction model. During the soil cracking process, decrease in potential energy due to the release of tensile stress is equal to the increase in surface energy because of increase in crack surface area:

$$\delta U\ge \delta {U}_{SE}$$

(44)

where $\delta U$ is the reduced potential energy and $\delta {U}_{SE}$ is the increased surface energy. The expression of critical stress based on Griffith energy criterion is:

$${\sigma }_{0}={\sigma }_{c}$$

(45)

where ${\sigma }_{0}$ is the fracture driving force; ${\sigma }_{c}$ is the fracture resistance, ${\sigma }_{c}=\sqrt{4\eta Y/\pi a,}$ $\eta$ is the surface energy per unit area, Y is the Young’s modulus.

Soil desiccation cracking is more in accordance with the crack mode I Morris et al. (1992) proposed a model that combines crack depth, the soil mechanical properties and matric suction profile to predict the depth of the cracks and onset of soil secondary cracking:

$$\begin{cases}{c}{K}_{I}^{1}=-0.4385\left(\frac{v}{1-v}{\gamma }_{d}+\frac{1-2v}{1-v}\frac{{S}_{0}}{h}\right){A}^{1.5}\\ {K}_{I}^{2}=1.1215\left(\frac{v}{1-v}{\gamma }_{d}A-\frac{1-2v}{1-v}{S}_{0}\left(1-\frac{A}{h}\right)\right){A}^{0.5}\\ {K}_{I}^{3}=\left(\frac{1-2v}{1-v}\left({S}_{0}\left(1-\frac{A}{h}\right)+{\gamma }_{w}(A-h)\right)+\frac{v}{1-v}\left({\gamma }_{s}-{\gamma }_{d}\right)(A-h)\right)g\left(\frac{Z}{A}\right)\end{cases}$$

(46)

where ${\text{S}}_{0}$ is the surface suction; h is the depth to water table; A is the half length of internal crack; Z is the depth variable; ${\gamma }_{d}$ is the unit weight of pore water, ${\gamma }_{s}$ is the unit weight of saturated soil, ${\gamma }_{d}$ is the unit weight of dry soil. In order to get the crack depth under critical conditions, fracture toughness K_IC should be determined when the crack initiated and developed in the soil:

$${K}_{IC}=\frac{-2\xi {E}_{c}}{1-{\nu }^{2}}$$

(47)

where $\xi$ is the specific surface energy, ${E}_{c}$ is the compression modulus.

Konrad and Ayad (1997b) pointed out that the ultimate crack depth depends on the tensile stress distribution on both crack sides and the value of the stress intensity factor, which varies with the length of the crack. In order to use LEFM to predict the depth of soil cracks, two assumptions were made: the actual tensile stress distribution is linearly applied on both sides of the crack; and the stress intensity factor corresponds to the trapezoidal tensile stress distribution:

$${K}_{I.trap}={K}_{I.rect}-{K}_{I.tria}$$

(48)

where ${\text{K}}_{\text{ = 1 * ROMAN .trap}}$ is stress intensity factors that corresponds to a trapezoidal tensile stress distribution, ${\text{K}}_{\text{ = 1 * ROMAN .rect}}$ is the stress intensity factors that corresponds to a uniform tensile stress distribution, ${\text{K}}_{\text{ = 1 * ROMAN .tria}}$ stress intensity factors corresponding to a triangular tensile stress distribution. In the case of initial soil cracks under the premise of ignoring gravitational stresses, the ultimate crack depth can be expressed as (Lachenbruch 1961):

$${\alpha }_{1}\left(\frac{b}{a}\right){\sigma }_{t}\sqrt{a}-{\alpha }_{2}\left(\frac{b}{a}\right)\left({\sigma }_{t}-{\sigma }_{3}\left(b\right)\right)\sqrt{a}={K}_{IC}$$

(49)

where ${\sigma }_{t}$ is tensile strength, ${\sigma }_{3}$ is lateral total stress, b is the depth over which the tensile stress is applied to the crack walls, ${\alpha }_{1}$ and ${\alpha }_{2}$ are coefficients depending on the given values of a and b, as shown in Fig. 18.

Fig. 18

Stress intensity functions for uniform and linear stress distributions (Konrad and Ayad 1997b)

Lakshmikantha et al. (2012) performed laboratory experiments on samples of different thicknesses and sizes, and the results indicated that soil cracking has obvious sample size effects. Although the initiation of soil cracks can be explained by the effective stress theory of classical soil mechanics, the development and propagation of soil cracks appears to be driven by energy. An indirect method of measuring the total length of a crack on the basis of LEFM was adopted to estimate the average cracking stress ${\sigma }_{a}$

$${\sigma }_{a}=\sqrt{\frac{{G}_{IC}E}{\pi a}}$$

(50)

The size-effect law proposed by Bazant (1984) can be expressed as:

$${\sigma }_{a}=\psi {\sigma }_{t}{\left(1+\frac{d}{{\lambda }_{0}{d}_{0}}\right)}^{-0.5}$$

(51)

where d is the characteristic dimension of specimen, ${\text{d}}_{0}$ is a reference dimension, $\psi$ and ${\gamma }_{0}$ are parameters that can be determined from a linear regression analysis. The experiments performed by Lakshmikantha et al. (2012) could validated Eq. (51), thus proving that LEFM can be adequately accurate in explaining the process of development and propagation of soil desiccation cracks. After measuring the tensile strength and fracture toughness of the samples in previous published work, the following empirical relationship was proposed (Lakshmikantha et al. 2008, 2012; Prat et al. 2008):

$${K}_{IC}={\alpha }_{t}{\sigma }_{t}^{n}$$

(52)

where ${\alpha }_{1}$ and n are the fitting proportionality coefficient. However, many researchers have performed a series of experiments to study the relationship between tensile strength and fracture toughness and pointed out that the two parameters have a good linear fit as expressed in Eq. (53) (Bhagat 1985; Harison et al. 1994; Wang et al. 2007b, 2020; Cao 2018).

$${K}_{IC}={\alpha }_{t}{\sigma }_{t}$$

(53)

These experimental results and curve fitting results are shown in Fig. 19. These results indicate that although fracture toughness and tensile strength of different soils have high linear correlation, there are also large differences in the values of proportionality coefficient and the determination coefficient. These observations may be due to differences in soil type and testing method, which is same as the measurement of tensile strength (Trabelsi et al. 2018). More experiments should be performed to unravel the reason for observed difference.

Fig. 19

Relationships between fracture toughness and tensile strength

5.2 Elastic plastic fracture mechanics EPFM

As discussed in Subsection 5.1, LEFM-based research assumes that the process of soil failure is a thermodynamic equilibrium process. This implies that the mechanical energy applied to the fracturing soil is equivalent to the energy required for soil cracking. For the linear elastic model of the soil cracking, the increase in mechanical energy dU will be converted into the increase in internal energy of the soil dw and the w is elastic and fully recoverable, which can be defined as ${dU=dw}_{el}$ (elastic energy ${w}_{el}$). When the external stress is removed, the soil can return to its original state. However, the assumption of this model fails to take into account the fact that a large amount of unrecoverable energy (plastic energy${\text{w}}_{\text{pl}}$) in the soil cracking process is dissipated. The ${\text{w}}_{\text{pl}}$ is mainly used for plastic processes including the soil particles rearrangement, the soil inter-particle friction and the debonding of inter-particle bonds (Abu-Hejleh and Znidarčić 1995).

In the EPFM model developed by Hallett and Newson (2005), the external energy is mainly decomposed into two parts, recoverable elastic energy ${\text{w}}_{\text{el}}$ and irrecoverable plastic energy ${\text{w}}_{\text{pl}}$. When the crack propagates, sufficient energy is needed to break the soil inter-particle bonds at the crack tip:

$$dU=d{w}_{el}+d{w}_{pl}+d\Phi$$

(54)

Where $d\Phi$ is the energy required for the destruction of the soil structure $d\Phi =2\mathrm{B}{\gamma }_{0}da$, which depends on the specific surface energy ${\gamma }_{0}$ and the increased surface area 2Bda (the amount of crack growth da and the crack thickness B) because of soil cracking. Therefore, Eq. (54) can be transformed into:

$$\frac{d\left(U-{w}_{el}\right)}{Bda}=\frac{d\left({w}_{pl}+\Phi \right)}{Bda}$$

(55)

The energy source of soil crack propagation is the driving force of crack D, expressed as:

$$C\equiv \frac{d\left(U-{w}_{el}\right)}{Bda}$$

(56)

The energy sink of soil crack propagation is the energy dissipation rate C, expressed as:

$$D\equiv \frac{d\left({w}_{pl}+\Phi \right)}{Bda}$$

(57)

Once the soil cracks begin to develop, the energy source and sink must be equal, C = D.

In the EPFM model, the J integral is an important parameter. First introduced by Rice (1968), it can be conveniently used to express the potential energy change rate during soil crack propagation at different water contents. Being that the J integral can be applied to evaluate the change of elastic potential energy and plastic potential energy, it is reasonable to use this parameter to analyze the elastic–plastic transition during crack development. Costa et al. (2012, 2013b, 2016) proposed a new ring test method to determine the elastoplastic fracture behavior of soil from the form of J integral. Different from the fracture parameters measured by the load control test, the J integral value measured from the natural drying ring test, is more suitable for use in the analysis of soil desiccation cracking.

The LEFM model is applicable to the problem of soil tension crack under stress. Although there is a certain gap between some assumptions in the theoretical model and the actual situation, the predicted soil crack depth has a certain reference in theory (Miller et al. 2016). A major disadvantage of the LEFM model is that the soil is not a brittle and linear elastic material, and energy dissipation from other processes (such as elastic mismatch, inter-particle friction, and micro-cracks) may be substantial (Hallett et al. 1995). Another limitation is that the LEFM model only considers the propagation of an individual crack and ignores the interactions among multiple cracks (Konrad and Ayad 1997b). These limitations lead to discrepancies between the theoretical predictions and the observed cracks in the actual situation. The EPFM model is an improvement on the LEFM model through consideration of the plastic deformation processes and irreversible plastic energy dissipation during soil cracking, which are more in accordance with the actual situation. Researchers have not only put forward that the soil cracking should be a non-linear fracture behavior (Vo et al. 2017) and created a process zone for plastic energy dissipation (Kendall and Weihs 1992), but have also proposed a new ring for testing the J integral parameter that is used to evaluate the elastoplastic fracture characteristics of natural soil during desiccation cracking (Costa et al. 2012, 2013b, 2016). However, there is still the challenge of accurately quantifying the parameters needed for use of the EPFM model and differentiating the elastic processes form the plastic processes during soil cracking. At present, the development of most LEFM and EPFM models is limited to numerical simulations due to lack of relevant laboratory and field confirmatory experiments. So, there is not enough experimental data to verify the rationality of the models. However, the application of LEFM and EPFM theory to soil cracking research has a great significance in existing soil cracking theory and provides a theoretical basis for the development of related cracking models in the future.

6 Numerical simulations of soil cracking based on fracture mechanics

To avoid the shortcomings of experiments, numerical simulation methods have been one of the important ways of studying cracking of soils (Sanchez et al. 2014). Due to the complexity of cracking initiation and propagation processes in soils, the fundamental mechanisms of soil cracking are still under investigation. The accuracy and reliability of most numerical models are under question. Current numerical simulation methods of soil cracking that are based on fracture mechanics can be categorized into two major groups: LEFM and cohesive crack method.

Lee et al. (1988) proposed a finite element model based on LEFM to predict crack propagation in brittle soil. The model splits a single node into two different nodes to represent the separation of material on both sides of the soil crack. The critical energy release rate is used as the material constant for soil tension cracks (Fig. 20). It is worth pointing out that the model has been verified with two field cases, including a soft soil embankment and an excavated slope.

Fig. 20

Proposed crack propagation mechanism: (a) before crack increment; (b) after crack increment (Lee et al. 1988)

Konrad et al. (1997b) proposed a highly idealized analytical model based on LEFM for predicting the spacings among the soil desiccation cracks and describing the phenomenon of crack propagation. This model can be applied to the dry shrinkage cracking characteristics of simulated mud, natural consolidated soil and compacted clay. Its predictions are in agreement with soil cracking experiments performed in an Australian coal mine tailings deposit.

Juárez-Luna et al. (2014) used a two-dimensional fracture mechanics finite element program based on the LEFM theory, to simulate soil crack initiation and propagation, and verified the control effect of critical stress intensity factor on soil crack development. The crack depths obtained with the numerical model had good agreement with field geotechnical data from the Valley of Mexico.

However, some researchers have indicated that LEFM may be more suitable for brittle soils. LEFM assumes infinite tensile stress at the crack tip, which is not valid when the material has limited tensile strength and displays significant plastic behavior around the crack process zone (Amarasiri and Kodikara 2011). The cohesive crack model is an attractive model for plastic soils that incorporate the main physical features of soil, particularly, fracture energy requirements and nonlinearity. Therefore, the cohesive crack method is more appropriate than LEFM for analyses of cracking of high moisture content soils due to the likely occurrence of the plasticity cracking zone (Amarisiri et al. 2014). The basic principle of the cohesive crack model is that the bridging stress σ during the opening the two crack faces, is a function of the crack opening near the crack tip (called the fracture process zone (FPZ) (Carpinteri and Colombo 1989)). The bridging stresses of a cohesive crack is shown in Fig. 21. This relationship between the bridging stress σ and the cohesive crack opening displacement w can be expressed as shown in Eq. (58).

Fig. 21

Bridging stresses of cohesive crack (Amarasiri and Kodikara 2011)

$$\sigma = \text{f} \left(w\right)$$

(58)

As the relative displacement between the two soil crack faces increases from zero to w, the crack bridging tensile stress gradually decreases from the tensile strength σ_t to zero at the same time.

Amarasiri and Kodikara (2011, 2013a, b) proposed a methodology that is based on cohesive crack method for numerical simulation of soil cracking. This mode does not have the requirement that an initial notch must be present and the bulk material must behave in a linear elastic manner. This method has been validated in some experiments, including a tension test with a rectangular cross section, a laboratory linearly constrained desiccation test, a notched three-point bend beam, a restrained ring, test a compact tension test and a field desiccation test performed in Saint Alban, Canada (Amarasiri and Kodikara, 2011, 2013a, b; Amarisiri et al. 2014).

Gui et al. (2016) developed a mix-mode cohesive fracture model for simulating soil desiccation cracking by employing a hybrid continuum-discrete method, implemented using Universal Distinct Element Code (UDEC). The cohesive fracture model considers tension, compression and shearing behavior of bonds among soil grains. The model produces results that are consistent with laboratory test results and deal with multiple fracture and deformation problems. The simulation results of three final crack patterns with different soil thickness (4 mm, 8 mm, and 16 mm) are shown in Fig. 22.

Fig. 22

The simulation results with different soil thickness. (a) 4 mm, (b) 8 mm, (c) 16 mm. All the three simulations with same soil-base strength, i.e., cohesion = 1.4 kPa, tensile strength = 0.7 kPa and friction angle = 5° (Gui et al. 2016)

Vo et al. (2017) built a cohesive damage-plasticity model to simulate the desiccation cracking of a clayey soil with consideration of soil hydraulic and mechanical behavior. This model can capture the initiation and propagation of soil cracks. Their results indicate that the model can reproduce trends observed in experiments (soil sample shrinkage, crack initiation and propagation, as shown in Fig. 23), and also enable investigation of the evolution of cracking soil mechanical parameters (stress, strain) and hydric state (suction, degree of saturation) at different soil specimen locations. In order to predict soil crack spacing and depth, a finite element code that enables incorporation of cohesive joint elements was used by Vo et al. (2019). From the results of numerical simulations, empirical correlations were developed to predict soil crack spacings and depths as functions of the surface suction on top of the soil, soil physical parameters and moisture evaporation rate. These results prove the validity of models with in situ experimental observations.

Fig. 23

(a) Description of the cracks observed at various moments and (b) distribution of horizontal displacement (Vo et al. 2017)

The cohesive zone model (CZM) was initially built to describe the plastic-zone beyond the crack tip (Barenblatt 1959). Pouya et al. (2019) developed the CZM by using finite element code to correctly predict the crack initiation and the ultimate crack depth. The numerical analysis shows that the maximum dissipated energy per unit crack depth can predict the soil desiccation crack depth and its related spacing. The CZM were successfully used to reproduce soil crack evolution and strain localization changes in the different fracture ductility clay beams flexural test (Hu et al. 2020).

Therefore, numerical models that are based on cohesive crack method can been valuable tools in soil cracking modeling with laboratory and field desiccation tests. The LEFM have been developed to study the propagation of a single crack. It can successfully predict crack depth from the soil suction and other mechanical parameters. However, the LEFM may be more suitable for brittle soils, rather plastic soils (Hallett and Newson 2005). On the other hand, the cohesive crack model is an attractive model for plastic soils because it incorporates the main physical features of a soil, including fracture energy and material nonlinearity. It is noted that the cohesive crack method is a calculation method based on EPFM that is widely used in the study of crack propagation (Ozenc et al. 2014). Meanwhile, the cohesive crack method has strong adaptability and can solve many nonlinear and large deformation problems. Compared with the stress intensity factor, the existence of cohesion keeps the crack tip to maintain a closing tendency, which can reduce or eliminate the stress singularity to a certain extent (van der Meer et al. 2012). Although the numerical simulation of soil cracking based on fracture mechanics can effectively predict the depth and spacing of soil desiccation cracks, the limited validation and verifications with field data as well as soil inhomogeneity make it necessary to test them more before wide application in geotechnical practice.

7 Conclusions and future works

This review synthesizes the past research efforts devoted to the experimental investigations and applications of fracture mechanics in soil cracking. This section presents concluding remarks and some outlooks requiring further investigations.

Five main test methods have been used to analyze soil fracture processes and parameters. It is obvious that each method has its advantages and limitations. Although the three-point bending test and the four-point bending test are very convenient methods of measuring soil fracture parameters, the self-weight of soil samples will affect the accuracy of the fracture parameters. While the compact tension test can overcome the influence of sample self-weight on the test results, failure of the soil loading point tends to precede the soil sample fracture because of the low tensile strength of most soils. The ring test is mainly used to obtain the value of J integral on the basis of EPFM, which has been widely applied in stress analyses. However, the ring test is a complicated measurement method when applied to soil cracking, and the test conditions are relatively strict, thereby rendering it unamenable to frequent use. The compression test is mainly applicable to frozen soils that exhibit higher strength. Pre-existing cracks and external loading of the compression samples influence the accuracy of fracture parameters of soil specimens with lower strength. Adjustment and standardization of test procedures, and development of easy-to-operate test devices are urgently needed for measurements of soil fracture parameters. This would enhance the convenience of replication of experiment for comparison, validation and verification of results of modeling.

The LEFM model is applicable to the problem of soil tension cracking under stress. Although there is a certain gap between some assumptions in the theoretical model and the actual situation, the predicted soil crack depth has reasonable rationale in theory. A major disadvantage of the LEFM model is that the soil is not a brittle and linear-elastic material, and energy dissipation from other processes lead to discrepancies between the theoretical predictions and experimental observations. The EPFM model is an improvement on the LEFM model through considerations of soil plasticity and irreversible plastic energy dissipation during soil cracking. These considerations are more reflective of the actual situation of soil cracking. However, it is still a challenge to accurately quantify the parameters that are incorporated into the EPFM model and distinguish between the elastic and plastic deformation processes during soil cracking. At present, the validation and verifications of most LEFM and EPFM models is limited to numerical simulations. Lack of relevant laboratory and field confirmatory experiments is a problem. However, the application of LEFM and EPFM theory in soil cracking research has high utility as regards the provision of the theoretical basis for the development of related cracking models in the future for field application.

Classical fracture mechanics is mainly applicable to brittle materials, while soil will show two states of brittleness and plasticity with the change of water content. Therefore, EPFM will be a key factor in using fracture mechanics to propose soil cracking models. Although fracture mechanics has made some progress in explaining soil crack propagation, there are still shortcomings in accurately judging the initiation time and position of soil crack. Hence, it is necessary to combine soil mechanics and fracture mechanics to complete a theoretical model to explain soil crack initiation and propagation. It is significant to fully understand the connections between soil microstructure, plasticity and cracking. The current research on fracture mechanics in soil cracking is mainly limited to laboratory tests, while the research results of field tests and numerical simulations are obviously insufficient. Although in-situ tests have problems such as time-consuming, laborious and high cost, such tests have an irreplaceable effect on the study of the essential laws of soil cracking. In addition, so far, the soil microstructure is rarely used in the research of soil fracturing. How to establish a connection between the quantitative parameters of the microstructure and the soil fracture parameters will be a challenging topic in the future.