1 Introduction

Breakage in granular materials has attracted significant attention from civil and mining engineers around the world regarding its profound impacts on the mechanical behavior of granular soils, including the reduction in maximum shear strength (Indraratna et al. 2015; Wang et al. 2013), an increase in creep and settlement of the structures located on these soils (Alonso et al. 2005; Leung et al. 1997), altering the dilative behavior of the soils into a more contractive response (Xiao et al. 2014; Charles and Watts 1980; Chen and Zhang 2016; Knittel et al. 2023), resulting in the occurrence of rapid long-runout motion of landslides (Zhu et al. 2022), changes in the critical state properties such as the critical friction angle and the critical void ratio (Bandini and Coop 2011; Ghafghazi et al. 2014; Kikumoto et al. 2010) as well as the permeability of granular materials (Fragaszy and Voss 1986; Papamichos et al. 1993). To reproduce the potential consequence(s) of the breakage process in finite element simulations, it is crucial to develop reliable constitutive models that incorporate the effects of particle breakage on the granular soil response. These constitutive models require an appropriate breakage index to estimate the degree of particle breakage occurring throughout the loading process. The ideal breakage index should not only precisely measure the progression of particle crushing, but also its formulation should allow a convenient incorporation into numerical modelling frameworks as part of constitutive models.

A wide variety of breakage indices have been introduced by researchers based on a comparison of pre- and post-test GSD curves determined by conventional sieve analysis (Leslie 1963; Lee and Farhoomand 1967; Lade et al. 1996; Hardin 1985; Einav 2007), micro-structure parameters including coefficient of uniformity, roundness and sphericity (Mehta and Patel (2018)). While these indices enable a precise estimation of particle breakage evolution, there is a significant limitation in incorporating them into constitutive models, particularly concerning the tracking of GSD evolution. This limitation stems from the fact that only a limited number of constitutive models include GSD as an input parameter in their formulation (Einav 2007). This prompts the requirement of proposing different formulations of breakage index, suitable to the specificity of each family of constitutive models, that can be easily implemented and later on tested for boundary value problems.

The main objective of this study is to develop breakage indices that could be conveniently incorporated into various constitutive models, including hypoplastic, elastoplastic, and thermodynamically consistent frameworks. For the first step, the role of the breakage index among constitutive equations of different types is discussed. One of the main criteria for developing an appropriate breakage index should be the balance between a reasonable number of parameters and the ease of computational implementation. In light of this, novel breakage indices that could be implemented in different constitutive models are introduced. The degree of particle breakage estimated using the proposed indices is compared against experimental data collected from monotonic and cyclic loading tests reported in the literature to validate their performance. The findings indicate that the suggested indices provide reasonable estimations of particle breakage under different loading conditions, and their formulation is convenient for future incorporation into advanced constitutive models.

2 The Role of Breakage Index in Constitutive Models

Various constitutive models have been specially developed to simulate the crushable soil response which can be mainly categorized into hyperplastic approaches [e.g., (Alaei et al. 2022, hypoplastic [e.g., (Qian et al. 2023; Engin et al. 2014], hypoelastic–plastic [e.g., (Xiao and Liu 2017)] and hyperelastic–plastic [(e.g., (Irani et al. 2022, 2023)] frameworks. Among the available constitutive models for crushable soils, those based on the breakage mechanic theory proposed by Einav (2007) are thermodynamically consistent. In this approach, the elastic stiffness tensor is derived based on a Helmholtz free energy function, expressed as a function of strain invariants and breakage index. Thereby, the breakage index plays a fundamental role in deriving elastic constitutive equations in thermodynamically admissible frames. A breakage-induced energy function is assumed here to provide an analytical explanation, and the corresponding fourth-order stiffness tensor is calculated. Geotechnical engineers typically prefer to initialize the stiffness tensor based on stress as it is easier to determine the natural in-situ stress state rather than the corresponding initial strain state. Thereby, a Gibbs energy function \(\psi\) in terms of Isomorphic stress invariants \(P^\prime\) and Q, and a breakage index B is assumed as follows:

$$\begin{aligned} \psi = (1-\nu B) \cdot \psi _r \end{aligned}$$
(1)

where \(\psi _r\) is the energy function formulated in terms of stress invariants \(P^\prime\) and Q for zero particle breakage and \(\nu\) is a material parameter. Here geotechnical sign convention is used (i.e. compression is positive). \(Q = \Vert \varvec{\sigma }^{\prime *} \Vert = \sqrt{\varvec{\sigma }^{\prime *}:\varvec{\sigma }^{\prime *}}\) and \(P^\prime = \text{tr}\,\varvec{\sigma }^\prime / 3\) are the deviatoric and mean stress Isomorphic invariants (where distances and inclinations are correctly represented) of the effective Cauchy stress tensor \(\varvec{\sigma }^\prime\), wherein the deviatoric part of stress tensor \(\varvec{\sigma }^{\prime *} =\varvec{\sigma ^\prime } - (\textrm{tr}\hspace{1pt}\varvec{\sigma ^\prime } / 3 ) ~{{\textbf {1}}}\) and for the double contraction (colon product) \({\textbf {A}}: {\textbf {B}} = {A}_{ij} {B}_{ij}\) holds. \(\text{tr}\,\varvec{\sigma }^\prime\) is the trace of the stress tensor and \(~{{\textbf {1}}}\) is the second-order identity tensor. The second derivative of the energy function in Eq. 1 yields the forth-order compliance tensor (\(\textsf{C}\)) as follows:

$$\begin{aligned} \textsf{C}= & {} \dfrac{\partial ^2\psi }{\partial \varvec{\sigma }^\prime \partial \varvec{\sigma }^\prime } = (1-\nu B) \dfrac{\partial ^2 \psi _r }{\partial \varvec{\sigma }^\prime \partial \varvec{\sigma }^\prime } + 2 \dfrac{\partial (1-\nu B)}{\partial \varvec{\sigma }^\prime } \dfrac{\partial \psi _r}{\partial \varvec{\sigma }^\prime } \nonumber \\{} & {} + \dfrac{\partial ^2 (1 - \nu B)}{\partial \varvec{\sigma }^\prime \partial \varvec{\sigma }^\prime } \psi _r \end{aligned}$$
(2)

One advantage of hyperelastic approaches is the one-to-one relationship between stress and elastic strain tensors that can be achieved through the Legendre transformation. Calculating the complementary energy function (e.g., switching from a Helmholtz energy potential to a Gibbs complementary one via the Legendre transformation) may require significantly more mathematical effort when the breakage index incorporates a combination of strain and stress tensor invariants. Hence, a breakage index expressed only as a function of the stress or strain invariants individually is favourable for hyperelastic as well as hyperelastic–plastic approaches. In addition, using a breakage index independent of the stress invariants simplifies the constitutive equations since this index would not participate in the first and second derivatives of the Gibbs energy potential, i.e., the last two terms in Eq. (2) would become zero. Nevertheless, formulating B as a function of either stress or strain components represents a significant advantage for its incorporation in any advanced constitutive models, especially in determining Jacobian and the time integration of rate equations. For instance, considering the energy function \(\psi\) in Eq. (1) results in the determination of the elastic strain tensor \(\varvec{\varepsilon }^e\) and its corresponding rate \({\dot{\varvec{\varepsilon }}}^e\) as follows:

$$\begin{aligned} \varvec{\varepsilon }^e = \dfrac{\partial \psi }{\partial \varvec{\sigma }^\prime } \; \; ; \; \; {\dot{\varvec{\varepsilon }}}^e = \dfrac{\partial ^2 \psi }{\partial \varvec{\sigma }^\prime \partial \varvec{\sigma }^\prime } {\dot{\varvec{\sigma }}^\prime} + \dfrac{\partial ^2 \psi }{\partial \varvec{\sigma }^\prime \partial B} {\dot{B}} \end{aligned}$$
(3)

where the superposed dot denotes the material time derivative. According to Eq.( 3), calculation of the rate of breakage index is required in the thermodynamically consistent frameworks. Moreover, adopting a breakage index independent of the stress invariants simplifies the formulation because the second term in Eq. (3) becomes zero.

A number of constitutive models have been introduced to simulate the mechanical behavior of crushable soils through the explicit incorporation of breakage indices into their constitutive equations (Indraratna and Salim 2002; Daouadji and Hicher 2010), or reformulation of the critical state line in the void ratio-mean effective stress-breakage index space (Hu et al. 2011; Xiao et al. 2014). For example, Hu et al. (2011) incorporated the breakage index into the description of the critical state line using the following equation:

$$\begin{aligned} e_c = e_\textrm{ref}(B) - \lambda \ln \left( {\dfrac{p^\prime }{p_\textrm{ref}} }\right) \end{aligned}$$
(4)

where \(e_c\) and \(p^\prime =~{{\textbf {1}}}:\frac{\varvec{\sigma }^\prime }{3}\) are the critical state void ratio and mean effective stress, respectively. \(p_\textrm{ref}\) and \(\lambda\) are material constants. \(e_\textrm{ref}\), the reference void ratio at \(p^\prime =p_\textrm{ref}\), varies as a function of the breakage index. The same concept has been used by Irani et al. (2022). Some constitutive models (e.g., (Liu et al. 2014; Cheng et al. 2022)) incorporate the energy dissipation caused by particle breakage into the traditional stress-dilatancy equation of Rowe (1962). The general formulation for the modified stress-dilatancy equation considering breakage is expressed as (Ueng and Chen 2000):

$$\begin{aligned} \dfrac{\sigma ^\prime _1}{\sigma ^\prime _3} = \left( 1 + \dfrac{{\textrm{d}} \varepsilon _v }{{\textrm{d}} \varepsilon _1} \right) \tan ^2{\left( 45+\frac{\varphi _\mu }{2} \right) } + \dfrac{1}{\sigma _3^\prime } \dfrac{{\textrm{d}} E_B}{{\textrm{d}} \varepsilon _1} (1 + \sin {\varphi _\mu }) \end{aligned}$$
(5)

where \(\sigma ^\prime _1\) and \(\sigma ^\prime _3\) are effective major and minor principal stress acting on triaxial specimens; \(\textrm{d}\varepsilon _v\) is the volumetric strain increment (positive in dilation) and \(\textrm{d}\varepsilon _1\) is the major principal strain increment (positive in compression); \(\varphi _\mu\) is the friction angle between soil particles; \(E_B\) is the energy dissipated due to particle breakage per unit volume of sand. Some of the constitutive models accounting for breakage incorporate Eq. (5) among their equations (e.g., (Liu et al. 2014), (Cheng et al. 2022)). Determining the energy consumption caused by particle breakage (\(E_B\)) is a challenging task that relies on physical measurements and the need for an assumption regarding the shape of the particles (Ueng and Chen 2000; Cheng et al. 2022). However, an alternative approach is to utilize an incremental breakage index, \(\textrm{d}B\), and calculate \(E_B\) as follows:

$$\begin{aligned} \text {d}{E_B} =\eta \cdot \text {d} B \end{aligned}$$
(6)

where \(\eta =q/p^\prime\) is the stress ratio, whereas \(q= \sqrt{\frac{3}{2}}\Vert \varvec{\sigma ^\prime }^* \Vert\) denotes the deviatoric stress. The role of the breakage index in the derivation of constitutive equations cannot be overstated. Hence, it is necessary to identify and utilize appropriate breakage indices for various constitutive models to ensure the validity, precision and numerical stability of the constitutive equation.

3 Development of New Shear-Induced Breakage Index Equations

Employing the breakage index for both compression and shear loading modes requires constitutive models with additional state or memory variables or even two yield surfaces. Indeed, some researchers reported negligible particle breakage after isotropic compression compared to the shear mode (Jia et al. 2017; Shahnazari and Rezvani 2013). Therefore, the ability to quantify particle breakage under shear loading is the main focus of the indices proposed in this article. It is widely recognized that particle breakage will eventually cease as soil samples undergo significant deformations or experience high levels of stress in the same direction (Coop et al. 2004; Xiao et al. 2014a) i.e., the breakage index will converge to a specific value, and the GSD will reach an endpoint (Einav 2007). Hence, the equations describing the extent of breakage should reach an asymptotic limit value. Moreover, any advanced constitutive model should be formulated in a fully tensorial form, with a reasonable number of parameters, incorporating the relevant state variables. Consequently, in the scope of this study, the formulation of new breakage indices that can be coupled in the future with advanced constitutive models should bear in mind the need to incorporate fewer (up to two) material constants, among other requirements. Regarding Eq. (3), the calculation of \({\dot{B}}\) is required for the time integration of rate equations in any constitutive model. Therefore, the corresponding breakage rate is calculated for each proposed breakage index. Breakage is irreversible (Guo and Zhu 2017), meaning that \({\dot{B}}\ge 0\) must hold. In the implementation of any constitutive model with breakage, users should be aware of controlling cases where \({\dot{B}} < 0\) by using Macaulay’s brackets \(\langle {\dot{B}} \rangle\) to calculate the breakage rate. The subsequent subsections introduce and discuss three types of breakage indices based on plastic work definition and individual stress and strain invariants. Material parameters for the proposed breakage indices were calibrated based on the compilation of reported tests on crushable soils, and the values are summarized in Table 2.

In this study, the breakage index proposed by Einav (2007) has been used to quantify particle breakage based on the GSD reported for experiments, interpreted as the ratio of the area between the initial and the current grading (\(B_t\)) over the area between the initial and the ultimate grading (\(B_p\)):

$$\begin{aligned} B = \dfrac{B_{t}}{B_{p}} = \dfrac{ \int \limits _{D_{m}}^{D_{M}} [F(D)-F_0(D)] \textrm{d}(\log (D)) }{ \int \limits _{D_{m}}^{D_{M}} [F_u(D) - F_0(D)] \textrm{d}(\log (D)) } \end{aligned}$$
(7)

wherein \(D_{m}\) is the minimum and \(D_{M}\) is the maximum grain diameter. \(F_0(D)\), F(D), and \(F_u(D)\) stand for the initial, current, and ultimate GSDs, respectively. Breakage index starts from zero value and evolves towards the final value of one, where GSD reaches its ultimate fractal distribution (Einav 2007). Hence, the desired equations for capturing the breakage extent should be bounded between the initial condition \(B=0\) and the maximum breakage \(B=1\). In the following sections, three new classes of breakage indices are formulated based on plastic shear work, plastic shear strain and stress invariants.

3.1 Breakage Index Based on Plastic Shear Work

The plastic work in granular materials mainly consists of the friction energy consumed by particle sliding, the work generated by dilatancy, and the energy consumed by particle breakage (Salim and Indraratna 2004; Einav 2007; Salimi and Lashkari 2020; Irani and Ghasemi 2020, 2021). During particle breakage upon loading, the contacts among particles change, significantly influencing the friction between them (Tarantino and Hyde 2005; Lobo-Guerrero and Vallejo 2005). Moreover, the progression of particle breakage affects the dilative behavior of crushable granular soils (Xiao et al. 2016a; Yu 2018). In this sense, particle breakage and plastic work are strongly interrelated. Many researchers used plastic work (\(W^{p}\)) for quantifying the breakage amount, whether by using an exponential model Xiao and Liu (2017) or through a hyperbolic equation, as those proposed by Daouadji et al. (2001) and Liu and Zou (2013):

$$\begin{aligned} B = \dfrac{W^{p}}{n+W^{p}} \; \; ; \; \; {W^{p}} = \int { \left\langle p^\prime ~{d\varepsilon }_v^p + q~{{d\varepsilon }_q^p} \right\rangle } \end{aligned}$$
(8)

where n is a constant, \({\varepsilon _v^p}\) and \({\varepsilon _q^p}\) denote the plastic volumetric and shear strain components, respectively. \(\langle x \rangle = x\) for \(x > 0\), and 0 otherwise. The Macaulay’s brackets were added by Hu et al. (2018) to enhance the prediction accuracy of the accumulation of particle breakage under cyclic loading or under loading paths where reversals in stresses or strains occur. In order to evaluate the performance of a breakage index expressed as a function of plastic work (see Eq. 8), a series of triaxial tests performed by Xiao et al. (2016b) on Tacheng rockfill have been employed. Deposits of Tacheng rockfill are found in the Shangrila County of China, containing mostly well-graded gravels. Table 1 illustrates the physical characteristics of the Tacheng rockfill. Xiao et al. (2016a) performed a series of large triaxial tests (specimens with 30 cm in diameter and 60 cm in height) with initial confining pressures of \(p^\prime _0=\) 0.2, 0.4, 0.8 and 1.6 MPa on loose, medium dense and dense specimens.

As seen in Fig. 1a, employing the established definition of breakage based on plastic work (Eq. 8) overestimates the grain-crushing extent of loose specimens. Considering loose and dense samples under the same applied load level reveals that more plastic work is dissipated in a loose assembly due to more significant volumetric changes than the one observed on dense specimens. Therefore, the breakage index calculated with Eq. (8) for a loose sample yields larger values, which conflicts with the experimental results observed for the Tacheng rockfill.

A solution to address the breakage overestimation is proposed in this research and consists of replacing the plastic work \((W^p)\) with the plastic shear work \((W^p_q)\) as follows:

$$\begin{aligned} B = \dfrac{W^p_{q}}{ c + W^p_{q}} \; \; ; \; \; {W^p_{q}}=\int \langle q{ \textrm{d}{\varepsilon }_q^p} \rangle \end{aligned}$$
(9)

wherein c is a new material parameter. Macaulay’s brackets are introduced to make the equation capable of calculating the evolution of particle breakage under cyclic loading. By relating the breakage index to \(W^p_{q}\) using a hyperbolic relationship (see Fig. 1b), particle breakage can be estimated more precisely for loose samples. However, Eq. (9) does not incorporate the effect of initial void ratio \((e_0)\), and the breakage extent cannot be estimated accurately due to the scattered experimental data observed for the Tacheng rockfill. The void ratio represents the available space for particles to slide and adjust the granular skeleton, which affects the evolution of particle breakage and should be considered in the quantification of particle breakage. In order to incorporate the effect of the initial void ratio \(e_0\), the new following breakage index is proposed:

$$\begin{aligned} B = \dfrac{{W^p_{q}}/{e_{0}}}{d+{W^p_{q}}/{e_{0}}} \; \; ; \; \; {\dot{B}} = \dfrac{\partial B}{\partial W^p_{q}} {\dot{W}}^p_{q} \end{aligned}$$
(10)

where d is a material constant. The performance of this equation for the Tacheng rockfill is presented in Fig. 1c. It is clear from the observation of Fig. 1c that the estimated particle breakage conforms precisely to the experimental data, irrespective of the initial relative density indicated by different \(e_0\) values or the applied confining pressures.

Fig. 1
figure 1

The evolution of particle breakage for the Tacheng rockfill versus a: plastic work \(W^{p}\), b: plastic shear work \(W^p_{q}\) and c: plastic shear work normalized with the initial void ratio \(W^p_{q}/e_{0}\) (experiments performed by (Xiao et al. 2016b))

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3.2 Breakage Index as a Function of Strain Tensor Invariants

Even though the newly proposed description of B using Eq. (10) is formulated in terms of stress and strain tensors invariants and could be incorporated into hypoplastic or hypoelastic–plastic constitutive models, it has some limitations for its implementation in thermodynamically consistent models. As discussed in Sect. 2, if the breakage index solely depends on stress or strain invariants, then using the Legendre transformation to determine the complementary free energy can be simplified. This study employs the plastic shear strain and void ratio to develop a new equation for a breakage index. The void ratio describes the volumetric deformation experienced by a granular assembly, whereas the shear-induced evolution of particle breakage until reaching the maximum value of \(B=1\) can be quantified using the plastic shear strain. The following equation is proposed in terms of void ratio (e) and plastic shear strain \((\varepsilon ^p_{q})\):

$$\begin{aligned} B = {1-\exp \left[ {-m \left( \dfrac{\varepsilon ^p_{q}}{e}\right) ^2} \right] } \; \; ; \; \; {\dot{B}} = \dfrac{\partial B}{\partial \varepsilon ^p_{q}} {\dot{\varepsilon }}^p_{q} + \dfrac{\partial B}{\partial e} {\dot{e}} \end{aligned}$$
(11)

where m is a material constant. Fig. 2-a illustrates the performance of Eq. (11) in predicting particle breakage of Tacheng rockfill. Equation (11) may be appropriate to be implemented into free energies which are functions of strain invariants.

Fig. 2
figure 2

The evolution of particle breakage for the Tacheng rockfill versus a strain invariant \(\varepsilon ^p_{q}/{e}\) and b stress invariant \({\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}\) (experiments performed by (Xiao et al. 2016b))

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3.3 Breakage Indices as a Function of Stress Tensor Invariants

The extension of any constitutive model from 2D conditions (usually formulated in the triaxial \(p^\prime -q\) space) into a full tensorial version should render the formulation more general and robust enough to cover as many loading/stress conditions as possible. General stress conditions involve all stress components or three stress invariants. Hence, the Lode angle can be used to generalize the formulation as the quantity representing the third invariant of the stress tensor in any arbitrary free energy function. Therefore, a new particle breakage is proposed as a function of mobilized friction angle, which directly depends on the three stress invariants: (Argyris et al. 1974; Tafili et al. 2021; Lashkari et al. 2022):

$$\begin{aligned} \sin {\left( \varphi _\textrm{mob}\right) } = \dfrac{3\eta }{6+\eta \cos {\left( 3\theta \right) }} \end{aligned}$$
(12)

wherein \(\eta\) represents the stress ratio, \(\varphi _\textrm{mob}\), and \(\theta\) are the mobilized friction angle and Lode angle, respectively.

In order to quantify breakage, the following equation is proposed:

$$\begin{aligned} B = \dfrac{{\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}}{k+{\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}} \; \; ; \; \; {\dot{B}} = \dfrac{\partial B}{\partial \sin {\left( \varphi _{\textrm{mob}}\right) }} {\dot{\sin }}{\left( \varphi _{\textrm{mob}}\right) } \end{aligned}$$
(13)

where k is a constant and \(\varphi _{\textrm{mob}_M}\) is the maximum mobilized friction angle. The maximum mobilized friction angle (instead of only mobilized friction angle) is introduced to make Eq. (13) capable of capturing breakage extent for specimens under cyclic loading and samples exhibiting a strain-softening response. Figure 2b shows the approximation of the experimental data with the new breakage index proposed in Eq. (13).

4 Evaluation of the Proposed Breakage Indices

The performance of the three new proposed breakage indices is validated using experimental results published for four different sands. A series of experiments performed by Wei et al. (2021) on a South China calcareous sand; Hyodo et al. (2017), Wu et al. (2018) and Hyodo et al. (2002) on Aio sand; Russell and Khalili (2004) on Kurnell sand; and Yamamuro and Lade (1996) on Cambria sand are employed. Table 1 presents the physical characteristics of these granular soils. The material constants for each breakage index were obtained through curve fitting. The calibrated parameters are presented in Table 2.

Table 1 Basic physical parameters of the considered granular materials
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Table 2 Constants used for estimating the amount of particle breakage of Tacheng rockfill, Calcareous, Kurnell, Cambria and Aio sands
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4.1 Breakage Index of South China Calcareous Sand

Wei et al. (2021) collected calcareous sand from one of the islands located in South China, which contained 81.08% of CaCO\(_3\). Specimens with four different initial dry densities (ranging from 45% to 93%) were prepared using the air pluviation method. Samples were consolidated under \(p^\prime _0\) = 100, 200, 400 and 1200 kPa and sheared up to 20% axial strain under drained condition. Figure 3a–c illustrates the performance of the proposed formulations in estimating breakage index based on the reported triaxial tests. The breakage indices of the specimens ranged from 0.029 to 0.1241. The experimental data confirms that Eqs. (10), (11) and (13) can estimate particle breakage to a satisfactory extent (Fig. 3a–c).

Fig. 3
figure 3

The evolution of particle breakage of South China calcareous sand versus a \(W^p_{q}/e_{0}\), b \(\varepsilon ^p_{q}/{e}\) and c: \({\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}\) (experiments performed by (Wei et al. 2021))

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4.2 Breakage Index of Aio Sand

Triaxial experiments performed by Hyodo et al. (2017), Wu et al. (2018) and Hyodo et al. (2002) have been employed to validate the introduced breakage indices under monotonic and cyclic conditions. The Aio sand considered in these experiments has sub-angular to angular grains and is found in the Yamaguchi prefecture southwest of Honshu Island of Japan. A series of drained monotonic and undrained cyclic tests were performed by Hyodo et al. (2017) with initial confining pressures of 0.1, 3 and 5 MPa using dense and medium-dense specimens. One set of constants has been employed to validate the proposed equations for the breakage index under both monotonic and cyclic loading. Reasonable consistency can be seen in Fig. 4 between the predicted amount of breakage and experimental results. In order to account for cyclic loading under triaxial extension and triaxial compression, the absolute value of plastic shear strain magnitude has been used in Fig. 4b.

Fig. 4
figure 4

The evolution of particle breakage of Aio sand versus \(W^p_{q}/e_{0}\), b \(\varepsilon ^p_{q}/{e}\) and c \({\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}\) (experiments performed by (Hyodo et al. 2017; Wu et al. 2018; Hyodo et al. 2002))

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4.3 Breakage Indices of Cambria and Kurnell Sand

In order to examine the particle breakage of quartz sand, experiments conducted on Kurnell sand have been considered. Russell and Khalili (2004) performed a series of triaxial tests on this sand under mean initial confining pressures of \(p^\prime _0\) = 0.76, 1.41, 2.39, 3.0, 5.7 and 7.8 MPa and with initial void ratios varying between 0.615 and 0.661. The drained triaxial tests were terminated at strain levels close to 40%. In addition, the test data from Yamamuro and Lade (1996) on Cambria sand is utilized to assess the performance of the proposed indices in estimating the degree of particle breakage under relatively high-pressure tests. Yamamuro and Lade (1996) employed the dry pluviation method for specimen preparation, and the tests were conducted in drained triaxial compression with a confining pressure of \(p^\prime _0=\) 2.1, 4, 5.8, 8.0, 11.5 and 15.0 MPa. All tests were carried out at the same initial void ratio of 0.52. Figure 5 shows the good performance of the breakage index formulations using Eqs. (10)–(13) for Cambria as well as Kurnell sand.

Fig. 5
figure 5

The evolution of particle breakage of Cambria and Kurnell sand versus a \(W^p_{q}/e_{0}\), b \(\varepsilon ^p_{q}/{e}\) and c: \({\sin {\left( \varphi _{\textrm{mob}_M}\right) }p^\prime _{0}}/{e_{0}}\) (experiments performed by (Yamamuro and Lade 1996; Russell and Khalili 2004))

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In general, the criteria for selecting an appropriate breakage index vary across different families of constitutive models and should consistently strike a balance between the minimum required material constants and the ease of computational implementation. Equation (13) is applicable for future developments in constitutive theory, which aims to conserve energy within a framework based on a free energy function expressed in terms of stress invariants, such as the Gibbs energy potential. Equation (10) is recommended for hypoelastic–plastic models, whereas Eq. (11) is in particular well suited for thermodynamically consistent frames with an energy function in terms of strain (e.g., Helmholtz energy function) as well as hypoelastic and hypoplastic models. For hypoplastic models, the incorporation of breakage indices based on an equivalent definition of plastic work and plastic strain invariants is possible considering the irreversible strain accumulated after an infinite small stress loop as plastic strain (Wu and Niemunis 1996).

The incorporation of the proposed and analyzed breakage indices in this paper introduces new parameters to the models. For the calibration of these parameters, at least two triaxial tests at different stress levels are required (which is in line with the standard for most advanced constitutive models). However, it must be ensured that grain crushing occurs during the shearing. Therefore, one might need to repeat one of the tests if the stress level is too low to cause the breakage of grains. The breakage indices involving plastic work demand a higher level of experience from the user because the data must be pre-processed in detail to calculate the plastic work.

Although numerous laboratory tests have been conducted to investigate the process of particle breakage in granular soils, limited knowledge is available regarding the evolution of particle crushing in triaxial extension mode, as well as under general and anisotropic loading paths following different Lode angles. Performing a series of true-triaxial and hollow cylinder tests may provide valuable insights for testing the proposed indices and developing new breakage indices. Additionally, the mechanical response of soils may be profoundly influenced by the in-situ stress state acting on the soil elements. However, the majority of studies have focused on isotropically consolidated samples, even though the actual in-situ stress state in the field is mainly anisotropic. This information can also be useful for evaluating constitutive models that consider the impact of the Lode angle on crushable soil behavior and the memory surface for triaxial tests under pre-sheared loading conditions. In tandem with this, the effect of particle crushing on the permanent deformations caused by high-cyclic loading (repeated loading with a large number of cycles \(N>10^3\) and a relatively small strain amplitude \(\varepsilon ^\text {ampl}<10^{-3}\) (Niemunis et al. 2005) remains unclear. Therefore, there is a need for future research projects to assess the ability of current breakage indices to estimate the extent of breakage under high cyclic loading.

5 Summary and Conclusion

This study aimed to determine the amount of breakage under shear loading and develop a convenient mathematical formulation. Three novel breakage indices have been suggested for future integration into constitutive models (such as hypoelastic–plastic, hypoplastic, or thermodynamically admissible frameworks) for crushable materials. The proposed indices have been evaluated using experimental data published in the literature for five different rockfill, calcareous and quartz granular materials. The novel indices provided estimated breakage values that were in satisfactory agreement with the available experimental data. The magnitude of the applied stress, void ratio, plastic shear strain level, plastic shear work, and mobilized friction angle were suggested as the appropriate variables to quantify the breakage amount in a soil assembly. The advantages and drawbacks of each breakage index have been discussed in detail, and recommendations for future incorporation into constitutive models have been given. For hypoplastic or hypoelastic–plastic models, all suggested indices and their rates are applicable. However, only those solely based on strain or stress invariants are considered adequate for energy-conserving frameworks. It would be desirable to have more experimental evidence covering a wider range of the variables employed to quantify the breakage extent and validate their performance for other crushable granular soils.