1 Introduction

The stability of excavated tunnels remains one of the most important and difficult problems in geotechnical engineering. To reveal the collapse mechanism of tunnel roof is of high significance for providing reference for the design and construction of tunnels. However, geotechnical materials are filled with cracks and fractures, which make geotechnical materials random variability of mechanical properties and far more complex than traditional elastic–plastic materials. Researchers have developed a variety of analytical approaches to investigate the tunnel roof stability and reveal the collapse mechanism of tunnel roof. Analytical approaches mainly include the limit analysis methods (e.g., Atkinson and Potts 1977; Davis et al. 1980; Yang and Yang 2010; Wang et al. 2014; Sloan and Assadi 1993; Lyamin and Sloan 2000; Osman et al. 2006; Klar et al. 2007; Fraldi and Guarracino 2009, 2010, 2011, 2012; Yang and Huang 2011; Yang and Yao 2017; Huang and Yang 2011) and the catastrophe theory (e.g., Zhang et al. 2014a, 2016; Zhang and Han 2015; Yang et al. 2017).

Most of analytical approaches mentioned above mainly focused on the first collapse mechanism of tunnel roof; however, the phenomena of progressive failures are also very common and important in geotechnical engineering (e.g., Stone and Wood 1992; Santichaianant 2002; Costa et al. 2009; Jacobsz 2016; Zhang et al. 2014b). For example, some typical failure mechanisms including the development of failure patterns in active trapdoor model tests are shown in Fig. 1 (e.g., Stone and Wood 1992; Santichaianant 2002; Costa et al. 2009; Jacobsz 2016). With the vertical movement of the trapdoor, first failure surface initiates from the corners of the trapdoor and propagates toward the center of the trapdoor. Then, continued vertical movement of the trapdoor would lead to the developments of a series of new failure surfaces. In the final stage involving relatively large trapdoor movements, the failure surface is almost vertical. Figure 2 shows the progressive failure in roof collapse of deep and shallow tunnels in model tests (Zhang et al. 2014b). The plane strain model tests were conducted to investigate the failure modes and dynamic evolutionary rules of soft ground tunnels in urban areas under two different cover depths (2.0D or 3.5D and D is tunnel diameter). As shown in Fig. 2a, b, the tunnel roof collapses under two different cover depths which are both progressive failure processes. Several collapses occur and at the same time several short-term stable collapsing arches form during each collapse. The failure under the condition of 3.5D does not reach the ground surface while the failure under the condition of 2.0D affects the ground surface. Although many researches were conducted using the numerical simulations and the model tests, there are few analytical researches on the progressive failure of tunnel roof after the initial collapse for the complexity of this phenomenon.

Fig. 1
figure 1

Progressive failure in active trapdoor model tests (e.g., Stone and Wood 1992; Santichaianant 2002; Costa et al. 2009; Jacobsz 2016)

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Fig. 2
figure 2

Progressive failure of tunnel roof collapse in model tests: a progressive failure of surrounding rock under the condition of 2D depth; b progressive failure of surrounding rock under the condition of 3.5D depth (Zhang et al. 2014b)

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Based on the above results of model tests, there are some features in progressive failure. First, the parameters characterizing the geotechnical materials are constantly changing and geotechnical materials undergo stress redistribution during the whole failure process. Second, failure surface propagation switches from one discontinuity to the next in a relatively sudden manner, which means that each collapsing block can be regarded as rigid. In this part, soil remains elastic and the total soil mass have displacement. Finally, soil dilatancy angle ψ defined as the vertical and the tangent along surface decreases from its initial value to zero and successive failure surfaces are in accordance with several discrete values of dilatancy angle ψ during the development of the successive failure surfaces (Stone and Wood 1992; Santichaianant 2002). In general, the phenomenon of progressive failure can be described as the continuous changes of system characteristic variables leading to several catastrophe changes of system state, which is fitting with the conception of the nature of the catastrophe theory (Thom 1972; Zeeman 1976; Arnold and Afraimovich 1999; Du 1994). Therefore, the catastrophe theory is well suited to analyze this kind of phenomenon.

Using the functional catastrophe theory, the first failure collapse mechanisms was studied and the first possible shapes of the collapsing blocks for deep and shallow tunnels was derived, respectively (Zhang et al. 2014a; Zhang and Han 2015). In this paper, functional catastrophe theory is also used to investigate the progressive collapse mechanisms and collapsing block shapes of circular tunnels under conditions of plane strain. The analytical solutions for the shape curve of the collapsing block of deep circular tunnels are derived based on the nonlinear power-law failure criterion and the non-associated flow rule. Moreover, criterions on progressive failure occurrence for deep tunnels are derived. Then, the analytical predictions obtained in this paper are compared with the corresponding experimental testing results.

2 Problem Description

2.1 Progressive Failure Mechanism of the Tunnel Roof

The estimation of the roof stability of deep tunnels primarily lies in determining the shape and dimension of the collapsing blocks which can actually collapse from the roof of the tunnel. To solve the proposed problem using the catastrophe theory, some assumptions are made. In this study, only the gravity field is considered, regardless of the tectonic stress field. The behavior of the rock mass is elastic-perfectly plastic. For the rock mass that follows the Hoek–Brown failure criterion, plastic potential energy within the total external load potential energy on the detaching surface is important and more dominant than elastic potential energy. Therefore, the paper ignores elastic potential energy and mainly concentrates on the plastic potential energy of the total external load potential energy on the detaching surface. The changes in the geometry of the collapsing block can be regarded as insignificant through the onset of the collapse (rigid-plastic behavior), which is consistent with the results of experimental tests (e.g., Stone and Wood 1992; Santichaianant 2002; Costa et al. 2009; Jacobsz 2016; Zhang et al. 2014b). The problem is considered as under conditions of plane strain. Progressive failure in roof collapse of deep tunnel model tests shown in Fig. 2 is emphatically analyzed in this paper. Therefore, progressive failure mechanisms of a tunnel roof adopted in this paper is shown in Fig. 3, in which L1 and h1 are the half-width and the intercept in y axis of the first collapsing block and L2 and h2 are the half-width and the intercept in y axis of the second collapsing block, respectively. R is the tunnel radius, ρ is the weight per unit volume of the rock mass, w is the thickness of the plastic detaching zone, and g(x) is a known function describing the shape of a circular tunnel:

$$g(x) = - \sqrt {R^{2} - x^{2} } .$$
(1)
Fig. 3
figure 3

Progressive failure mechanisms of the deep tunnel roof: a first collapsing block of a tunnel roof; b possible successive collapsing block of a tunnel roof

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The first consideration adopted in this paper is that the following derivation is based on the nonlinear failure criterion and the non-associated flow rule. Many experiments have shown that the failure envelop of soils is not linear in the σn–τn stress space, and the linear failure criterion is merely a particular case. Thus, a nonlinear failure criterion may be more suitable for the stability analysis of geotechnical structures. For this reason, the nonlinear power-law failure criterion is adopted in this paper, which can be expressed as follows (Zhang and Chen 1987; Zhang and Wang 2015; Yang and Yao 2017):

$$\tau_{\text{n}} = c_{0} \left( {\frac{{\sigma_{\text{n}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{1/m} ,$$
(2)

where c0 = initial cohesion of soil at zero stress; σn and τn = normal and shear stresses on the failure surface, respectively; σt = absolute value of tensile stress when τ is equal to zero; and m = nonlinear coefficient.

Using the non-associated flow rule, the real deformation and failure characteristics of soil can be better simulated. Similar to the associated flow rule, the velocity at velocity discontinuities for a soil following a non-associated flow rule inclines at an angle, dilatancy angle ψ, with respect to the velocity discontinuity line. In general, the dilatancy angle ψ varies from zero to the friction angle φ (0 ≤ ψ ≤ φ). Correspondingly, dilative coefficient, η, which relates the dilatancy angle and the soil friction angle, is defined as:

$$\eta = \frac{\psi }{\varphi }.$$
(3)

Theoretically, the magnitude of dilative coefficient is 0 ≤ η ≤ 1. The case η = 1 indicates that the material follows an associated flow rule. According to reference (Zhang and Wang 2015), when the geotechnical materials subject to nonlinear power-law failure criterion and non-associated flow rule, Eq. (2) can be modified to Eq. (4). For the sake of derivation, the term ηc0 is replaced by a new parameter c0i:

$$\tau_{{{\text{n}}i}} = \eta_{i} c_{0} \left( {\frac{{\sigma_{{{\text{n}}i}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{1/m} = c_{0i} \left( {\frac{{\sigma_{{{\text{n}}i}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{1/m} .$$
(4)

The second key consideration adopted in this paper is that dilatancy angle ψ decreases from its initial value to zero and successive failure surfaces are in accordance with several discrete values of dilatancy angle ψ during the development of the successive failure surfaces. In addition, it should be pointed that the dilatancy angle ψ can be considered as a function of plastic strain and confining stress, and subsequently a dilatancy factor Kψ that decays from an initial value Kψ in accordance with an exponential function of plastic shear strain is proposed as follows (Detournay 1986; Alejano and Alonso 2005):

$$K_{\psi } = 1 + \left( {K_{{\psi ,{\text{peak}}}} - 1} \right)e^{{ - {{\gamma^{\text{p}} } \mathord{\left/ {\vphantom {{\gamma^{\text{p}} } {\gamma^{{{\text{p}}*}} }}} \right. \kern-0pt} {\gamma^{{{\text{p}}*}} }}}} ,$$
(5)

where Kψ = (1 + sinψ)/(1 − sinψ).

By making simple derivation, the variation of dilatancy angle ψ along with of plastic shear strain γp is as follows:

$$\sin \psi = \frac{{\sin \psi_{0} }}{{\sin \psi_{0} + \left( {1 - \sin \psi_{0} } \right)e^{{{{\gamma^{p} } \mathord{\left/ {\vphantom {{\gamma^{p} } {\gamma^{{{\text{p}}*}} }}} \right. \kern-0pt} {\gamma^{{{\text{p}}*}} }}}} }}\quad \left( {\psi \;{\text{from}}\;\psi_{0} \;{\text{to}}\;0} \right).$$
(6)

As shown in Eq. (6), dynamic (continuous) variation of plastic shear strain γp leads to dynamic (continuous) variation of dilatancy angle ψ, which results in dynamic (continuous) variation of dilative coefficient η in Eq. (4). However, based on the features in progressive failure mentioned above in Introduction, not the whole continuous values of dilatancy angle ψ correspond to the tunnel roof collapses but only several discrete values of dilatancy angle (ψ1, ψ2, ψ3, …) correspond to several collapses. Therefore, the catastrophe state analysis of progressive failure lies in the derivation of the occurrence conditions for each collapse and determining the shape and dimension of the collapsing blocks.

The failure mechanism of tunnel roof (Fraldi and Guarracino 2009) is modified by considering variable detaching velocity along yield surface (Li and Yang 2017). And they concluded that the impact of variable detaching velocity is more remarkable on the shape of collapsing block. Therefore, the third key consideration adopted in this paper is the variable detaching velocity along detaching surface to obtain more consistent prediction results with the corresponding experimental testing results.

2.2 Catastrophe Theory and Catastrophic Conditions for a Functional

Since the deformation and failure of tunnel surrounding rock are characterized by nonlinear and discontinuous phenomena, catastrophe theory is suitable for investigating the behaviors of tunnel collapse. The nature of the catastrophe theory is continuous change of system characteristic quantity which leads to catastrophe change of system state. The results of model test show that several continuous collapses of tunnel roof correspond to several discrete value of dilatancy angle. Moreover, the dilatancy angle changes continuously with plastic shear strain. In other words, progressive failure can be considered as successive catastrophe phenomena, which means that continuous change of plastic shear strain γp leads to several catastrophe changes of system state.

If the potential function of the system is defined by a functional J [f(x)], as in Eq. (7), determining the non-Morse critical point fc(x) of the potential function of the system becomes challenging:

$$J\left[ {f(x)} \right] = \int\limits_{a}^{b} {F\left[ {x,\,f(x),\,f^{\prime}\left( x \right)} \right]dx} ,$$
(7)

where the primes indicate the derivatives of the functions with respect to their subscript coordinates, i.e., \(f^{\prime}\left( x \right) = \partial f\left( x \right)/\partial x.\)

With a subsection integral, the two specific forms of the catastrophic conditions are obtained for functional J [y] as follows (Zhang et al. 2014a; Zhang and Han 2015):

$$\frac{\partial F}{\partial f\left( x \right)} - \frac{\partial }{\partial x}\left( {\frac{\partial F}{{\partial f^{\prime}\left( x \right)}}} \right) = 0,\;({\text{Euler equation}}),$$
(8)
$$\frac{{\partial^{2} F}}{{\partial f\left( x \right)^{2} }} - 2\frac{\partial }{\partial x}\left( {\frac{{\partial^{2} F}}{{\partial f\left( x \right)\partial f^{\prime}\left( x \right)}}} \right) + \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {\frac{{\partial^{2} F}}{{\partial f^{\prime}\left( x \right)^{2} }}} \right) = 0,\;({\text{Catastrophic Euler equation}}).$$
(9)

3 Catastrophe State Analysis of a Deep Tunnel Collapse

3.1 First Collapsing Block of a Tunnel Roof

This paper proposed a new approach to determine critical state of tunnel roof collapse based on plastic strain energy and catastrophe theory. Based on the basic theory of elastic–plastic mechanics, the stress state of a point flows along the yield surface when the stress state of a point yields and the plastic strain energy of a point will increase. Therefore, the plastic strain energy can be used as an index to evaluate the state of the studied system. The larger the plastic strain is and the larger the accumulated plastic strain energy is. The collapse occurs when plastic strain energy reaches a certain value (i.e., critical state or catastrophic state).

To analyze the catastrophe state of a deep tunnel collapse, the total potential energy of the studied system should be first obtained. The total potential energy of the studied system includes the plastic strain energy of the internal forces on the detaching zone and the external force power induced by the weight of the detaching zone.

First, for standard geotechnical materials (i.e., those obeying to an associated flow rule), the plastic potential, Ψ, is assumed to be coincident with the nonlinear power-law yield curve and takes the following form by virtue of Eq. (4):

$$\varPsi = \tau_{{{\text{n}}1}} - c_{01} \left( {\frac{{\sigma_{{{\text{n}}1}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{1/m} .$$
(10)

So that the plastic strain rate can be derived as follows:

$$\dot{\varepsilon }_{\text{n}} = \lambda \frac{\partial \varPsi }{\partial \sigma } = - \lambda \frac{{c_{01} }}{{m\sigma_{\text{t}} }}\left( {\frac{{\sigma_{\text{n1}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{{\left( {1 - m} \right)/m}} ,$$
(11)
$$\dot{\gamma }_{\text{n}} = \lambda \frac{\partial \varPsi }{\partial \tau } = \lambda .$$
(12)

Moreover, from a purely geometrical line of reasoning, the plastic strain rate components can be written in the following form (Fraldi and Guarracino 2009):

$$\dot{\varepsilon }_{\text{n}} = \left( {\dot{u}/w} \right)\left[ {1 + f^{\prime}\left( x \right)^{2} } \right]^{ - 1/2} ,$$
(13)
$$\dot{\gamma }_{\text{n}} = - \left( {\dot{u}/w} \right)f^{\prime}\left( x \right)\left[ {1 + f^{\prime}\left( x \right)^{2} } \right]^{ - 1/2} .$$
(14)

The comparison of Eqs. (13) and (14) shows that:

$$\lambda = - \left( {\dot{u}/w} \right)f^{\prime}\left( x \right)\left[ {1 + f^{\prime}\left( x \right)^{2} } \right]^{ - 1/2} .$$
(15)

Then, substituting Eq. (15) into (11), it follows:

$$\dot{\varepsilon }_{\text{n}} = \left( {\dot{u}/w} \right)f^{\prime}\left( x \right)\left[ {1 + f^{\prime}\left( x \right)^{2} } \right]^{ - 1/2} \frac{{c_{01} }}{{m\sigma_{\text{t}} }}\left( {\frac{{\sigma_{\text{n1}} + \sigma_{\text{t}} }}{{\sigma_{\text{t}} }}} \right)^{{\left( {1 - m} \right)/m}} .$$
(16)

Considering the compatibility, the plastic strain rate components in Eqs. (13) and (16) must be equated, that is:

$$\sigma_{\text{n}} = \sigma_{\text{t}} \left( {\frac{{m\sigma_{\text{t}} }}{{f^{\prime}\left( x \right)c_{01} }}} \right)^{m/(1 - m)} - \sigma_{\text{t}} .$$
(17)

Combining Eqs. (17) and (4), the following equation is obtained:

$$\tau_{\text{n}} = c_{01} \left( {\frac{{m\sigma_{\text{t}} }}{{f^{\prime}\left( x \right)c_{01} }}} \right)^{1/(1 - m)} .$$
(18)

Now the plastic strain energy of the internal forces on the detaching zone is derived using Eqs. (13) and (14), and (17) and (18):

$$U_{i} = \int {\left( {\sigma_{\text{n}} \varepsilon_{\text{n}} + \tau_{\text{n}} \gamma_{\text{n}} } \right){\text{d}}u} = \left( {u/w} \right)\left[ {1 + f^{\prime}\left( x \right)^{2} } \right]^{ - 1/2} \left\{ { - \sigma_{\text{t}} + \sigma_{\text{t}}^{1/(1 - m)} \left( {\frac{{c_{01} }}{m}} \right)^{m/(m - 1)} \left[ {1 - m} \right]f^{\prime}\left( x \right)^{m/(m - 1)} } \right\}.$$
(19)

For the geotechnical materials that obey non-associated flow rule, the plastic strain energy of the internal forces on the detaching zone should be modified into the following form (Li and Yang 2017) by virtue of two parameters P and Q:

$$\begin{aligned} U_{i} & = \sigma_{\text{n}} \varepsilon_{\text{n}} + \tau_{\text{n}} \gamma_{\text{n}} \\ & = Q\left[ { - \sigma_{t} + \sigma_{t} \left[ {c_{01} /(m\sigma_{t} )} \right]^{m/(m - 1)} (1 - m)\left[ {\frac{P}{Q}f_{1}^{\prime } (x)} \right]^{m/(m - 1)} } \right] \times {u \mathord{\left/ {\vphantom {u {\left[ {w\sqrt {1 + f_{1}^{\prime } (x)^{2} } } \right]}}} \right. \kern-0pt} {\left[ {w\sqrt {1 + f_{1}^{\prime } (x)^{2} } } \right]}}, \\ \end{aligned}$$
(20)

where P and Q are two parameters describing the impacts of variable detaching velocity on the shape curves of collapsing blocks, which are defined as (η′ is the angle between u and the vertical direction) (Li and Yang 2017):

$$\left\{ {\begin{array}{*{20}l} {P = 1 + \frac{{\tan \eta^{\prime } }}{{f_{1}^{\prime } \left( x \right)}},} \hfill \\ {Q = 1 - \tan \eta^{\prime } f_{1}^{\prime } \left( x \right).} \hfill \\ \end{array} } \right.$$

In addition, the work of the applied loads per unit length (We) of the detaching surface is

$$W_{\text{e}} = \rho \left[ {f_{1} (x) - g(x)} \right]u.$$
(21)

Then, the total potential energy of the studied system can be expressed as:

$$J[x,f_{1} (x),f_{1}^{\prime } (x)] = - \int_{0}^{{L_{1} }} {\rho [f_{1} (x) - g(x)]u{\text{d}}x} + \int\limits_{0}^{{L_{1} }} {Q\left[ { - \sigma_{\text{t}} + \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} (1 - m)\left[ {\frac{P}{Q}f_{1}^{\prime } (x)} \right]^{m/(m - 1)} } \right]u{\text{d}}x} .$$
(22)

Based on the functional catastrophe theory, the function F studied is

$$F = F[x,f_{1} (x),f_{1}^{\prime } (x)] = \left\{ { - \rho [f_{1} (x) - g(x)] + Q\left[ { - \sigma_{\text{t}} + \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} (1 - m)\left[ {\frac{P}{Q}f_{1}^{\prime } (x)} \right]^{m/(m - 1)} } \right]} \right\}u.$$
(23)

The key in the catastrophic state analysis is to find the specific expression of f1(x) with the help of Eqs. (8) and (9).

Substituting Eq. (23) into Eqs. (8) and (9), the explicit forms of the group of differential equations of f1(x) for the problem are

$$- \rho - \frac{\text{d}}{{{\text{d}}x}}\left( { - Q\left( {\frac{P}{Q}} \right)^{m/(m - 1)} \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} mf_{1}^{\prime } (x)^{1/(m - 1)} } \right) = 0,$$
(24)
$$\frac{{{\text{d}}^{2} }}{{{\text{d}}x^{2} }}\left( { - Q\left( {\frac{P}{Q}} \right)^{m/(m - 1)} \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} \frac{m}{m - 1}f_{1}^{\prime } (x)^{{\left( {2 - m} \right)/(m - 1)}} } \right) = 0.$$
(25)

The detaching curve f1(x) is obtained by integrating Eq. (24):

$$f_{1} (x) = QP^{ - m} \left( {\sigma_{\text{t}} \rho^{m - 1} /c_{01}^{m} } \right)(x + \rho^{ - 1} \tau_{0} )^{m} - h_{1} ,$$
(26)

where τ0 and h1 are two integration constants.

Substituting Eq. (26) into (25), the following equation is obtained:

$$\frac{{P^{m} }}{Q}\left( {\frac{{\rho^{2 - m} c_{01}^{m} }}{{\sigma_{\text{t}} }}} \right)\left( {\frac{2 - m}{m}} \right)(x + \rho^{ - 1} \tau_{0} )^{ - m} = 0.$$
(27)

Equation (27) is required to be zero for any value of x. This means that the value of m must be 2, which is the result of Eq. (9), i.e., one of the catastrophic conditions. In addition, the parameter m determines the power exponent of f1(x), so it also describes the shape of the collapsing block. Based on the value of m, the reduced form of f1(x) can be obtained:

$$f_{1} (x) = QP^{ - 2} \left( {\sigma_{\text{t}} \rho /c_{01}^{2} } \right)(x + \rho^{ - 1} \tau_{0} )^{2} - h_{1} .$$
(28)

There are two unknown parameters τ0 and h1 in Eq. (28), which can be determined by transversality conditions in variational analysis. It states that Eq. (23) should satisfy both Eqs. (29) and (30) (Zhang et al. 2014a; Zhang and Han 2015):

$$\left. {\frac{\partial F}{{\partial f_{1}^{\prime } (x)}}} \right|_{x = 0} = 0,$$
(29)
$$F - \left[ {f_{1}^{\prime } (x) - g^{\prime}(x)} \right]\left. {\frac{\partial F}{{\partial f_{1}^{\prime } (x)}}} \right|_{{x = L_{1} }} = 0.$$
(30)

Substituting Eq. (23) into Eqs. (29) and (30), the explicit forms of the transversality conditions are

$$\begin{aligned} & - \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} mf_{1}^{\prime } (0)^{1/(m - 1)} \\ & \quad = - Q\left( {\frac{P}{Q}} \right)^{m/(m - 1)} \sigma_{t} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} m\left[ {m\left( {\sigma_{\text{t}} \rho^{m - 1} /c_{01}^{m} } \right)(0 + \rho^{ - 1} \tau_{0} )^{m - 1} } \right]^{1/(m - 1)} = 0, \\ \end{aligned}$$
(31)
$$\begin{aligned} & \left\{ { - \rho [f_{1} (L_{1} ) - g(L_{1} )] + Q\left[ { - \sigma_{\text{t}} + \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} (1 - m)\left[ {\frac{P}{Q}f_{1}^{\prime } (L_{1} )} \right]^{m/(m - 1)} } \right]} \right\} \\ & \quad - \left[ {f_{1}^{\prime } (L_{1} ) - g^{\prime}(L_{1} )} \right]\left[ { - Q\left( {\frac{P}{Q}} \right)^{m/(m - 1)} \sigma_{\text{t}} \left[ {c_{01} /(m\sigma_{\text{t}} )} \right]^{m/(m - 1)} mf_{1}^{\prime } (L_{1} )^{1/(m - 1)} } \right] = 0. \\ \end{aligned}$$
(32)

Simplifying Eqs. (31) and (32), the values of τ0 and h1 in the expression of collapsing block are determined as follows:

$$\tau_{0} = 0,$$
(33)
$$h_{1} = Q\left( {\sigma_{\text{t}} /\rho - g(L_{1} ) + g^{\prime}(L_{1} )L_{1} } \right).$$
(34)

Substituting the values of τ0 and h1 into Eq. (28), the shape curve of collapsing block can be obtained as follows:

$$f_{1} (x) = Q\left\{ {\left( {\sigma_{\text{t}} \rho /c_{01}^{2} } \right)\left( {\frac{x}{P}} \right)^{2} - \left[ {\sigma_{\text{t}} /\rho - g(L_{1} ) + g^{\prime}(L_{1} )L_{1} } \right]} \right\}.$$
(35)

Now we obtain the curve which describes the shape and dimensions of the collapsing block of deep tunnels. The curve is a parabola with y axis being the axis of symmetry and h1 being the y intercept.

However, the value of L1 is still unknown in Eq. (35). Then the value of L1 can be easily obtained using the geometric compatibility condition:

$$f_{1} (x = L_{1} ) = g(x = L_{1} ).$$
(36)

Substituting Eqs. (1) and (35) into Eq. (36) results in:

$$Q\left( {\sigma_{\text{t}} \rho /c_{01}^{2} } \right)\left( {\frac{{L_{1} }}{P}} \right)^{2} - Q\left( {\sigma_{\text{t}} /\rho } \right) + \left( {Q - 1} \right)g(L_{1} ) - Qg^{\prime}(L_{1} )L_{1} = 0.$$
(37)

From the mathematical perspective, whether or not the first collapse occurs is equivalent to whether Eq. (37) has a solution under the condition of 0 < L1 < R.

Moreover, it is possible to compute the overall weight of the first collapsing block per unit length (P1) by

$$P_{1} = 2\int\limits_{0}^{{L_{1} }} {\rho [g(x) - f_{1} (x)]{\text{d}}x} .$$
(38)

3.2 Possible Successive Collapsing Block of a Tunnel Roof

Similarly, the shape curves of possible successive collapsing blocks can be obtained as:

$$f_{i} (x) = Q\left\{ {\left( {\sigma_{\text{t}} \rho /c_{{0{\text{i}}}}^{2} } \right)\left( {\frac{x}{P}} \right)^{2} - \left[ {\sigma_{\text{t}} /\rho - f_{i - 1} (L_{i} ) + f_{i - 1}^{\prime } (L_{i} )L_{i} } \right]} \right\}.$$
(39)

Similarly, the value of Li is still unknown in Eq. (39). Then the value of L can be easily obtained using the geometric compatibility condition:

$$f_{i} (x = L_{i} ) = f_{i - 1} (x = L_{i} ).$$
(40)

Moreover, from the mathematical perspective, whether or not the successive collapse occurs is equivalent to whether Eq. (40) has a solution under the condition of 0 < Li < Li−1.

Substituting Eqs. (35) and (39) into Eq. (40) results in:

$$L_{i}^{2} = \frac{{P^{2} }}{{\rho^{2} }}\left( {\frac{{c_{0,i}^{2} c_{0,i - 1}^{2} }}{{c_{0,i - 1}^{2} - 2c_{0,i}^{2} }}} \right).$$
(41)

Considering the inequation 0 < Li, the result is as follows:

$$c_{0,i} < \frac{{c_{0,i - 1} }}{\sqrt 2 }\;{\text{or}}\;\eta_{i - 1} < \frac{\sqrt 2 }{2}.$$
(42)

In addition, considering the inequation Li<  Li−1, the result is as follows:

$$\eta_{i - 1} < \frac{{L_{i - 1} \rho }}{{\sqrt {P^{2} c_{0,i - 1}^{2} + 2L_{i - 1}^{2} \rho^{2} } }}.$$
(43)

With Eqs. (42) and (43), the condition of progressive failure occurrence for deep tunnel based on nonlinear power-law failure criterion is as follows:

$$\eta_{i - 1} < \hbox{min} \left( {\frac{\sqrt 2 }{2},\frac{{L_{i - 1} \rho }}{{\sqrt {P^{2} c_{0,i - 1}^{2} + 2L_{i - 1}^{2} \rho^{2} } }}} \right).$$
(44)

The occurrence of progressive failure in tunnel roofs from an analytical perspective was analyzed (Fraldi and Guarracino 2012) and it is demonstrated that within the framework of limit analysis after the collapse of a first block from the tunnel roof no additional detachments can take place (i.e., ηi−1 = 1). This shows those results in Eq. (34) are consistent with the conclusion (Fraldi and Guarracino 2012). As a consequence, it is further confirmed that the attention must be focused mainly on degradation processes of the rock mass.

Moreover, it is possible to compute the overall weight of each collapsing block per unit length (Pi) by

$$P_{i} = 2\int\limits_{0}^{{L_{i} }} {\rho [f_{i - 1} (x) - f_{i} (x)]} {\text{d}}x.$$
(45)

4 Comparisons with the Results of Model Test

The results obtained from this paper were compared with the results from model tests (Zhang et al. 2014b). The plane strain model tests were conducted to investigate the failure modes and dynamic evolutionary rules of soft ground tunnels in urban areas under different overburden depths. The geometrical similarity ratio of the model tests were 1/30. A kind of material, composed of barite, quartz and vaseline, was selected to represent the surrounding ground. The weight ratio of the ingredients barite:quartz:vaseline was 8.0:5.0:0.6. The tunnel excavation process was modeled by the pressure release of an airbag inside the tunnel. The stress field of the model test was produced by gravity alone.

The comparison between the estimated values by experimental and analytical results is shown in Fig. 4. The specific initial values of parameters in this model test are c01 = 20 kPa, σt = 22 kPa and ρ = 18 kN/m3, R = 3 m. Table 1 shows the specific values of parameters in each collapse of model test.

Fig. 4
figure 4

Comparisons of analytical results with the results obtained from model test: a experimental testing results; b analytical results (P = 1, Q = 1)

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Table 1 Values of parameters in each collapse of model test
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The results show that the modes and rules of progressive failure of deep circular tunnel are similar. The authors claim that the precise extraction of dynamic test parameters is very difficult which leads to the differences of experimental and analytical results. Moreover, as shown in Fig. 5, only the variation of plastic shear strain γp (or dilatancy angle ψ) is considered and the variations of other parameters are not considered in this paper, which also leads to the differences of experimental and analytical results.

Fig. 5
figure 5

Catastrophe state analysis of a deep tunnel collapse

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Furthermore, the impacts of variable detaching velocity on the shape curves of the several continuous collapsing blocks are shown in Fig. 6, which indicates that detaching velocity (P and Q) has a big effect on the shape curves of collapsing blocks. Therefore, variable detaching velocity should be considered to obtain more consistent prediction results with the corresponding experimental testing results.

Fig. 6
figure 6

Comparisons with two cases of P and Q: a first collapse; b second collapse; c final collapse (experimental testing results: blue curves; analytical results of P = 1 and Q = 1: red curves; analytical results of P = 1.2 and Q = 0.8: green curves) (color figure online)

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5 Conclusions

This paper uses functional catastrophe theory to investigate the progressive collapse mechanisms and collapsing block shapes of circular tunnels under conditions of plane strain. The main conclusions are as follows.

  1. 1.

    In general, the phenomenon of progressive failure can be described as the continuous changes of system characteristic variables leading to several catastrophe changes of system state, which is fitting with the conception of the nature of the catastrophe theory. Therefore, the catastrophe theory is well suited to analyze this kind of phenomenon and adopted in this paper.

  2. 2.

    The analytical solutions for the shape curve of the collapsing block of circular tunnels are derived based on the nonlinear power-law failure criterion and non-associated flow rule. Moreover, the conditions of progressive failure occurrence for deep tunnel were proposed. Those conditions would be used to judge whether or not the first and successive collapse occurs.

  3. 3.

    The analytical predictions obtained in this paper are compared with experimental testing results. The comparisons show that our analytical predictions are consistent with the corresponding experimental testing results, thus demonstrating the validity of the proposed analytical methodology.