Analytical Solutions for Tunnels of Elliptical Cross-Section in Rheological Rock Accounting for Sequential Excavation

H. N. Wang; S. Utili; M. J. Jiang; P. He

doi:10.1007/s00603-014-0685-7

Rock Mechanics and Rock Engineering

September 2015, Volume 48, Issue 5, pp 1997–2029 | Cite as

Analytical Solutions for Tunnels of Elliptical Cross-Section in Rheological Rock Accounting for Sequential Excavation

Authors
Authors and affiliations

H. N. WangEmail author
S. Utili
M. J. Jiang
P. He

Original Paper

First Online: 20 December 2014

Received: 30 March 2014

Accepted: 09 November 2014

Abstract

Time dependency in tunnel excavation is mainly due to the rheological properties of rock and sequential excavation. In this paper, analytical solutions for deeply buried tunnels with elliptical cross-section excavated in linear viscoelastic media are derived accounting for the process of sequential excavation. For this purpose, an extension of the principle of correspondence to solid media with time varying boundaries is formulated for the first time. An initial anisotropic stress field is assumed. To simulate realistically the process of tunnel excavation, solutions are developed for a time-dependent excavation process with the major and minor axes of the elliptical tunnel changing from zero until a final value according to time-dependent functions specified by the designers. In the paper, analytical expressions in integral form are obtained assuming the incompressible generalized Kelvin viscoelastic model for the rheology of the rock mass, with Maxwell and Kelvin models solved as particular cases. An extensive parametric analysis is then performed to investigate the effects of various excavation methods and excavation rates. Also the distribution of displacements and stresses in space at different times is illustrated. Several dimensionless charts for ease of use of practitioners are provided.

Keywords

Rheological rock Non-circular tunnel Analytical solution Sequential excavation

List of symbols

A, A₀ and A_i ($ i = 1,\;2, \ldots ,\;\infty $): Coefficients in inverse conformal mapping
$ A_{i}^{{j{\text{B}}}} $ and $ A_{ij}^{{k{\text{B}}}} $: Coefficients correlated to coordinates and material parameters of generalized Kelvin model in Appendix
$ A_{i}^{{j{\text{M}}}} $ and $ A_{ij}^{{k{\text{M}}}} $: Coefficients correlated to coordinates and material parameters of Maxwell model in Appendix
a(t): Function of half major axis with respect to time
a₀: Initial value of half major axis (at time t = 0)
a₁: Final value of half major axis
B_i ($ i = 1,\;2, \ldots ,\;9 $): Coefficients in displacement solutions
$ B_{i}^{j} $ ($ i = 1,\;2 $; $ j = 1,\;2 $): Terms defined in Eqs. (66) and (68)
b(t): Function of half minor axis with respect to time
b₀: Initial value of half minor axis (at time t = 0)
b₁: Final value of half minor axis
$ C_{i}^{\text{B}} $ ($ i = 1,\;2 $): Coefficients correlated to material parameters of generalized Kelvin model in Appendix
$ C_{1}^{\text{M}} $: Coefficients correlated to material parameters of Maxwell model in Appendix
c(t): Parameter in conformal mapping (defined in Eq. (26))
d: Ratio of major over minor axis
D_i ($ i = 1,\;2 $): Coefficients in stress solutions (in Eq. 37)
$ F_{i}^{j} $ ($ i = 1,\;2 $; $ j = 1,\;2 $): Terms defined in Eqs. (55), (56), (57) and (58)
f₀: Inverse conformal mapping with respect to variable z
f₁: Inverse conformal mapping with respect to variable z ₁
G(t): Time-dependent relaxation shear modulus for viscoelastic model
G_e: Shear modulus of elastic problem
G_H: Shear elastic modulus of the Hookean element in the Generalized Kelvin model
G_K: Shear elastic modulus of the Kelvin element in the Generalized Kelvin model
G_S: Permanent shear modulus of the generalized viscoelastic model: $ G_{\text{S}} = {{G_{\text{H}} G_{\text{K}} } \mathord{\left/ {\vphantom {{G_{\text{H}} G_{\text{K}} } {(G_{\text{H}} + G_{\text{K}} )}}} \right. \kern-0pt} {(G_{\text{H}} + G_{\text{K}} )}} $
H: Function defined in Eq. (48)
I₁ and I₂: Function defined in Eq. (59)
K(t): Time-dependent relaxation bulk modulus in the rock viscoelastic model
K_e: Bulk modulus of elastic problem
l: Number of items in inverse conformal mapping
m(t): Parameter in conformal mapping (defined in Eq. (26))
n_j: Vector indicating the direction normal to the boundary
$ (n_{\chi } ,\;n_{\tau } ) $: Local coordinates
$ n_{\text{r}}^{\text{K}} $: Normalized excavation rate for the generalized Kelvin model
$ n_{\text{r}}^{\text{M}} $: Normalized excavation rate for the Maxwell model
P_i ($ i = 1,\;2,\;3 $): Prescribed time-dependent stresses at stress boundary
P_x (P_y): Tractions (surface forces) along the x (y) direction on the stress boundary
p₀: Vertical compressive stress at infinity
p_x (p_y): Boundary tractions (surface forces) applied on the tunnel wall to calculate the excavation-induced displacements and stresses
q: Number of points adopted to determine the coefficients of inverse conformal mapping
R^*: Radius of the axisymmetric problem used to normalize displacements
$ S_{\sigma } $ ($ S_{u} $): Time-dependent stress (displacement) boundaries
s: Variable in the Laplace transform
$ s_{ij}^{\text{e}} $, $ e_{ij}^{\text{e}} $: Tensors of the stress and strain deviators for the elastic case
$ s_{ij}^{\text{v}} $, $ e_{ij}^{\text{v}} $: Tensors of the stress and strain deviators for the viscoelastic case
$ T_{\text{K}} $: Retardation time of Kelvin component of the generalized Kelvin viscoelastic model
$ T_{\text{M}} $: Relaxation time of the Maxwell viscoelastic model
t: Time variable (t = 0 is the beginning of excavation)
$ t_{1} $: End time of excavation
$ t{'} $: Time variable ($ t{'} $ = 0 is the time the initial pressure applied)
$ t_{0}^{'} $: Start time of excavation
$ u_{i}^{{}} $: Prescribed displacements on the displacement boundary
$ u_{x}^{{({\text{A}}){\text{v}}}} $($ u_{y}^{{({\text{A}}){\text{v}}}} $): Displacement corresponding to viscoelastic problem of case A (in Cartesian coordinates)
$ u_{x}^{{({\text{C}}){\text{v}}}} $($ u_{y}^{{({\text{C}}){\text{v}}}} $): Excavation-induced displacement for viscoelastic problem (in Cartesian coordinates)
$ u_{\chi }^{{({\text{C}}){\text{v}}}} $($ u_{\tau }^{{({\text{C}}){\text{v}}}} $): Excavation-induced displacement for viscoelastic problem (in local coordinates)
$ u_{x}^{\text{e}} $ ($ u_{y}^{\text{e}} $): Displacement along x (y) direction for the elastic problem
$ u_{x}^{\text{s}} $ ($ u_{y}^{\text{s}} $): Prescribed displacement along x and y direction on displacement boundary
$ u_{x}^{\text{v}} $ ($ u_{y}^{\text{v}} $): Displacement along x (y) direction for the viscoelastic problem
$ u_{i}^{\text{v}} $ ($ \sigma_{ij}^{\text{v}} $): Displacements (stresses) tensor for the viscoelastic problem
$ u_{s}^{\text{e}} $: Radial displacement at the tunnel wall for the axisymmetric elastic problem with radius $ R^{*} $ and shear modulus $ G_{\text{S}} $
$ u_{s0}^{\text{e}} $: Radial displacement at the tunnel wall for the axisymmetric elastic problem with radius $ R^{*} $ and shear modulus $ G_{\text{H}} $ in the Maxwell model
$ u_{i}^{*} $ ($ \sigma_{ij}^{*} $): Displacements (stresses) tensor obtained by replacing $ G_{\text{e}} $ with $ s\fancyscript{L}\left[ {G(t)} \right] $ and $ K_{\text{e}} $ with $ s\fancyscript{L}\left[ {K(t)} \right] $ in the general solution for the associated elastic problem
v_r: Cross-section excavation rate
X: Position vector of a point on the plane
X₀: Position vector of a point on the boundary
(x, y): Cartesian coordinates
z: Complex variable: z = x + iy
z_A: Arbitrary point on the boundary
z₀: Generic point on the time-dependent boundary at time $ t{'} $
z_σ: Point on time-dependent stress boundary
z₁: Complex variable defined in Eq. (32)
z_1j: Boundary points in z ₁ plane determined by Eq. (33) corresponding to point $ \zeta_{j}^{{}} $

Greek symbols

α: Angle between $ n_{\chi } $ and x direction
$ \delta $: Dirac delta function
$ \delta_{ij} $: Unit tensor
$ \gamma $: Function with respect to s obtained by replacing $ G_{\text{e}} $ with $ s\fancyscript{L}\left[ {G(t)} \right] $ and $ K_{\text{e}} $ with $ s\fancyscript{L}\left[ {K(t)} \right] $ in $ \kappa $
$ \Delta P_{x}^{{}} $($ \Delta P_{y}^{{}} $): Prescribed stresses along the boundaries in calculation of excavation-induced displacements and stresses
$ \Delta s_{ij}^{\text{v}} $ ($ \Delta e_{ij}^{\text{v}} $): Incremental stresses (strains) induced by tunnel excavation
$ \zeta $: Complex variable: $ \zeta = \xi + \eta i $
$ \zeta_{j}^{{}} $: Points in $ \zeta $ plane determined by Eq. (34) corresponding to point z ₁
$ \eta $: Imaginary part of $ \zeta $
$ \eta_{\text{K}} $: Viscosity coefficient of the dashpot element in the generalized Kelvin model
$ \kappa $: Material coefficient defined by Eq. (14)
$ \lambda $: Ratio of horizontal over vertical stress
$ \xi $: Real part of $ \zeta $
$ (\rho ,\;\theta ) $: Polar coordinates
$ \sigma_{ij}^{\text{v}} $ ($ \varepsilon_{ij}^{\text{v}} $): Stress (strain) tensor for viscoelastic case
$ \sigma_{kk}^{\text{e}} $ ($ \varepsilon_{kk}^{\text{e}} $): Mean stress (strain) for elastic case
$ \sigma_{kk}^{\text{v}} $ ($ \varepsilon_{kk}^{\text{v}} $): Mean stress (strain) for viscoelastic case
$ \sigma_{x}^{\text{v}} $, $ \sigma_{y}^{\text{v}} $: Normal stress along x and y direction for viscoelastic case
$ \sigma_{x}^{\text{e}} $, $ \sigma_{y}^{\text{e}} $: Normal stress along x and y direction for elastic case
$ \sigma_{xy}^{\text{v}} $ ($ \sigma_{xy}^{\text{e}} $): Shear stress for viscoelastic (elastic) case
$ \sigma_{x}^{{({\text{A}})}} $, $ \sigma_{y}^{{({\text{A}})}} $, $ \sigma_{xy}^{{({\text{A}})}} $: Stresses corresponding to viscoelastic problem of case A (in global Cartesian coordinates)
$ \sigma_{x}^{{({\text{C}})}} $, $ \sigma_{y}^{{({\text{C}})}} $, $ \sigma_{xy}^{{({\text{C}})}} $: Excavation-induced stresses (in global Cartesian coordinates)
$ \sigma_{\chi }^{{({\text{A}})}} $, $ \sigma_{\tau }^{{({\text{A}})}} $, $ \sigma_{\chi \tau }^{{({\text{A}})}} $: Stresses corresponding to viscoelastic problem of case A (in local Cartesian coordinates)
$ \sigma_{\chi }^{{({\text{C}})}} $, $ \sigma_{\tau }^{{({\text{C}})}} $, $ \sigma_{\chi \tau }^{{({\text{C}})}} $: Excavation-induced stresses (in local Cartesian coordinates)
$ \varphi_{1}^{{}} $ and $ \psi_{1}^{{}} $: Two complex potentials
$ \varphi_{2}^{{}} $ and $ \psi_{2}^{{}} $: Two complex potentials obtained by replacing $ G_{\text{e}} $ with $ s\fancyscript{L}\left[ {G(t)} \right] $ and $ K_{\text{e}} $ with $ s\fancyscript{L}\left[ {K(t)} \right] $ in $ \varphi_{1}^{{}} $ and $ \psi_{1}^{{}} $
$ \varphi_{1}^{{({\text{A}})}} $ and $ \psi_{1}^{{({\text{A}})}} $: Two complex potentials for the elastic problem A
$ \varphi_{1}^{{({\text{B}})}} $ and $ \psi_{1}^{{({\text{B}})}} $: Two complex potentials for the elastic problem B
$ \varphi_{1}^{{({\text{C}})}} $ and $ \psi_{1}^{{({\text{C}})}} $: Two complex potentials for calculating the excavation-induced displacements and stresses for the elastic case
$ \omega $: Conformal mapping determined in Eq. (25)

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Notes

Acknowledgments

This work is supported by National Basic Research Program of China (973 Program) with Grant No.2014CB046901; State Key Lab. Of Disaster Reduction in Civil Engineering (Grant No. SLDRCE14-B-11); Marie Curie Actions—International Research Staff Exchange Scheme (IRSES): GEO—geohazards and geomechanics with Grant No. 294976; China National Funds for Distinguished Young Scientists with Grant No. 51025932. These supports are greatly appreciated.

Appendix 1

This appendix presents the analytical derivation of the first two terms in Eqs. (20) and (21) for the purpose of calculating the excavation-induced stresses and displacements. The two Eqs. (20) and (21) are listed in the following as Eqs. (53) and (54):

$$ \begin{aligned} s_{ij}^{\text{v}} (t{'} ) = & 2e_{ij}^{\text{v}} (0^{ + } )G(t{'} ) + 2\int_{0}^{{t^{'}_{{0}^{ -} } }} {G(t{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ & \quad + 2\int_{{t^{'}_{{0}^{ +} } }}^{{t{'} }} {G(t{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau + 2\left[ {e_{ij}^{\text{v}} (t^{'}_{{0}^{ + }} ) - e_{ij}^{\text{v}} (t^{'}_{{0}^{ - }} )} \right]G\left( {t{'} - t^{'}_{0} } \right) \\ \end{aligned} $$

(53)

$$ s_{ij}^{\text{v}} (t^{'}_{0} ) = 2e_{ij}^{\text{v}} (0^{ + } )G(t^{'}_{0} ) + 2\int_{0}^{{t^{'}_{{0}^{ -} } }} {G(t^{'}_{0} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau . $$

(54)

The first and second terms in Eq. (53) are as follows:

$$ {\text{The first term:}}\quad F_{1}^{1} = 2e_{ij}^{\text{v}} (0^{ + } )G(t{'} ) $$

(55)

$$ {\text{The second term:}}\quad F_{2}^{1} = 2\int_{0}^{{t^{'}_{{0}^{ -} } }} {G(t{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau $$

(56)

The corresponding terms in Eq. (54) are as follows:

$$ {\text{The first term:}}\quad F_{1}^{2} = 2e_{ij}^{\text{v}} (0^{ + } )G(t_{0}^{'} ) $$

(57)

$$ {\text{The second term:}}\quad F_{2}^{2} = 2\int_{0}^{{t^{'}_{{0}^{ -} } }} {G(t_{0}^{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau $$

(58)

The expressions for displacements and strain rates in the rock before the excavation starts needed in the derivation, are obtained in Appendix 1.1. Note that the exact expressions of the coefficients are not given, since only the form of functions of the coefficients with respect to the given parameters are necessary in the demonstration.

1.1 Expressions for Displacements and Strain Rates Before the Excavation

Substituting Eq. (29) into Eq. (15), and assuming that:

$$ I_{\rm 1}(t{'} ) = \fancyscript{L}^{ - 1} \left[ {\frac{1}{{s\fancyscript{L}\left[ {G(t{'} )} \right]}}} \right],\quad I_{\rm 2}(t{'} ) = \fancyscript{L}^{ - 1} \left[ {\frac{{\kappa_{\text{L}} (s)}}{{s\fancyscript{L}\left[ {G(t{'} )} \right]}}} \right]. $$

(59)

The displacements occurred prior to the excavation can be calculated as (solution B-vis):

$$ u_{x}^{{({\text{B}}){\text{v}}}} + iu_{y}^{{({\text{B}}){\text{v}}}} = \frac{{(1 + \lambda )p_{0} z + 2(1 - \lambda )p_{0} \overline{z} }}{8}\int_{0}^{{t{'} }} {H(t{'} - \tau ){\text{d}}\tau } - \frac{{(1 + \lambda )p_{0} z}}{8}\int_{0}^{{t{'} }} {I(t{'} - \tau ){\text{d}}\tau } . $$

(60)

Substituting the functions expressing the shear and bulk relaxation moduli for the adopted viscoelastic model into Eq. (60), the explicit expressions can be obtained. If only shear viscoelasticity is considered, i.e., K(t) = K _e, displacements can be derived as follows:

$$ u_{i}^{{({\text{B}}){\text{v}}}} = A_{i}^{{1{\text{M}}}} (z) + A_{i}^{{2{\text{M}}}} (z)t{'} + A_{i}^{{3{\text{M}}}} (z)\exp ( - \lambda_{{{\text{M}}1}} t{'} ),\quad i = 1,\;2,\lambda_{{{\text{M}}1}} > 0,\quad {\text{for the Maxwell model}} $$

(61)

$$ \begin{aligned} u_{i}^{{({\text{B}}){\text{v}}}} = & A_{i}^{{1{\text{B}}}} (z) + A_{i}^{{2{\text{B}}}} (z)\exp ( - \lambda_{{{\text{B}}1}} t{'} ) + A_{i}^{{3{\text{B}}}} (z)\exp ( - \lambda_{{{\text{B}}2}} t{'} ), \\ & \quad i = 1,\;2,\;\lambda_{{{\text{B}}1}} > 0,\;\lambda_{{{\text{B}}2}} > 0 \quad {\text{for the generalized Kelvin model}} \\ \end{aligned} $$

(62)

where the terms with subscript i = 1 denote the components along the x direction, and i = 2 denote the components along the y direction. The coefficients $ A_{i}^{{j{\text{M}}}} $, $ A_{i}^{{j{\text{B}}}} $ (i = 1, 2 and j = 1–3) are determined via Eq. (60).

According to Eqs. (61) and (62), and the strain–displacement relationships, $ \varepsilon_{ij}^{\text{v}} = \frac{1}{2}[u_{i,j}^{\text{v}} + u_{j,i}^{\text{v}} ] $, as well as the definition of strain deviators given in Eq. (2), the derivative of strain deviators for time t’ can be calculated as follows:

$$ \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}t{'} }} = A_{ij}^{{4{\text{M}}}} (z) + A_{ij}^{{5{\text{M}}}} (z)\exp ( - \lambda_{{{\text{M}}1}} t{'} ),\;i = 1,\;2,\,j = 1,\;2 \quad {\text{for the Maxwell model}} $$

(63)

$$ \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}t{'} }} = A_{ij}^{{4{\text{B}}}} (z)\exp ( - \lambda_{{{\text{B}}1}} t{'} ) + A_{ij}^{{5{\text{B}}}} (z)\exp ( - \lambda_{{{\text{B}}2}} t{'} ),\;i = 1,\;2,\;j = 1,\;2\quad {\text{for the generalized Kelvin model}} $$

(64)

For models with unlimited viscosity (Type B models), e.g., Burgers model, the expressions for displacement and strain rate are analogous to Eqs. (61) and (63), where the values of the coefficients A to be used depend on the rheological models; conversely for models of limited viscosity, e.g., Kelvin model, the expressions are analogous to Eqs. (62) and (64).

1.2 Derivation for the generalized Kelvin model

According to the expression of G(t) for the generalized Kelvin model in Eq. (46), when time tends to infinity or is large enough, G(t) will be a constant, which is a general law for Type A viscoelastic models. Because the excavation started at a time much later than the time when the initial stresses were applied, the time $ t_{0}^{'} $ (beginning of excavation,), and the generic time $ t{'} \ge t^{'}_{0} $, can be assumed to be infinite. Therefore, $ G(t{'} ) = G(t_{0}^{'} ) $ and the first terms in Eqs. (53) and (54) are equal, that is $ F_{1}^{1} = F_{1}^{2} $.

Substituting Eq. (46) into Eq. (56), yields

$$ \begin{aligned} F_{2}^{1} &= 2\int_{0}^{{t^{'}_{{0}^{ - }} }} {G(t{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau = 2\int_{0}^{{t^{'}_{{0}^{ - }} }} {\left[ {C_{1}^{B} \exp \left[ { - \lambda_{\text{B}} (t{'} - \tau )} \right] + C_{2}^{B} } \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ &= 2\left\{ {\int_{0}^{{t^{'}_{{0}^{ -} } }} {C_{1}^{B} \exp \left[ { - \lambda_{B} (t{'} - \tau )} \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau + \int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{2}^{\text{B}} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau } } \right\} \\ &= 2\left( {B_{1}^{1} + B_{2}^{1} } \right), \\ \end{aligned} $$

(65)

where $ C_{1}^{B} $ and $ C_{2}^{B} $ are coefficients independent of time, and:

$$ \begin{aligned} B_{1}^{1} = & \int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{1}^{B} \exp \left[ { - \lambda_{B} (t{'} - \tau )} \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ B_{2}^{1} = & \int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{2}^{B} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau } . \\ \end{aligned} $$

(66)

Substituting Eq. (46) into Eq. (58), yields:

$$ \begin{aligned} F_{2}^{2} &= 2\int_{0}^{{t^{'}_{{0}^{ - }} }} {G(t_{0}^{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau = 2\int_{0}^{{t^{'}_{{0}^{ - }} }} {\left[ {C_{1}^{B} \exp \left[ { - \lambda_{B} (t_{0}^{'} - \tau )} \right] + C_{2}^{B} } \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\& = 2\left\{ {\int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{1}^{B} \exp \left[ { - \lambda_{B} (t_{0}^{'} - \tau )} \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau + \int_{0}^{{t^{'}_{{0}^{ -} } }} {C_{2}^{B} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau } } \right\} \\ &= 2\left( {B_{1}^{2} + B_{2}^{2} } \right), \\ \end{aligned} $$

(67)

where:

$$ \begin{aligned} B_{1}^{2} = & \int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{1}^{B} \exp \left[ { - \lambda_{B} (t_{0}{'} - \tau )} \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ B_{2}^{2} = & \int_{0}^{{t^{'}_{{0}^{ - } }}} {C_{2}^{B} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau } . \\ \end{aligned} $$

(68)

It can be noted from Eqs. (66) and (68) that $ B_{2}^{1} = B_{2}^{2} $. Substituting Eq. (64) into the expression of $ B_{1}^{1} $ (in Eq. 66), yields:

$$ B_{1}^{1} = \int_{0}^{{t^{'}_{{0}^{ - }} }} {C_{1}^{B} \exp \left[ { - \lambda_{B} (t{'} - \tau )} \right]} \left[ {A_{ij}^{4B} \exp ( - \lambda_{B1} \tau ) + A_{ij}^{5B} \exp ( - \lambda_{B2} \tau )} \right]{\text{d}}\tau . $$

(69)

After integration, then rearranging:

$$ \begin{aligned} B_{1}^{1} & = D_{ij}^{1B} \left[ {\exp \left( { - \lambda_{3} (t{'} - t_{0}^{'} )} \right)\exp \left( { - \lambda_{B1} t{'} } \right) - \exp \left( { - \lambda_{B} t{'} } \right)} \right] \\ & \quad + D_{ij}^{2B} \left[ {\exp \left( { - \lambda_{4} (t{'} - t_{0}^{'} )} \right)\exp \left( { - \lambda_{B2} t{'} } \right) - \exp \left( { - \lambda_{B} t{'} } \right)} \right], \\ \end{aligned} $$

(70)

where $ \lambda_{3} = \lambda_{B} - \lambda_{B1} > 0 $, $ \lambda_{4} = \lambda_{B} - \lambda_{B2} > 0 $. When $ t{'} \to \infty $ and $ t_{0}^{'} \to \infty $, Eq. (70) becomes:

$$ \left. {B_{1}^{1} } \right|_{\begin{subarray}{l} t{'} \to \infty \\ t_{0}^{'} \to \infty \end{subarray} } = 0 $$

(71)

Substituting Eq. (64) into the expression of $ B_{1}^{2} $ yields:

$$ B_{1}^{2} = D_{ij}^{1B} \left[ {\exp \left( { - \lambda_{B1} t_{0}^{'} } \right) - \exp \left( { - \lambda_{B} t_{0}^{'} } \right)} \right] + D_{ij}^{2B} \left[ {\exp \left( { - \lambda_{B2} t_{0}^{'} } \right) - \exp \left( { - \lambda_{B} t_{0}^{'} } \right)} \right] $$

(72)

when $ t_{0}^{'} \to \infty $, $ B_{1}^{2} = 0 $. According to Eqs. (65) and (67), as well as the fact that $ B_{2}^{1} = B_{2}^{2} $, $ B_{1}^{1} = B_{1}^{2} = 0 $, the second term of Eq. (53) is equal to that of Eq. (54), that is, $ F_{2}^{1} = F_{2}^{2} $. Owing to the fact that the first and second term in Eq. (53) are equal to the corresponding terms in Eq. (54), the equality between the sums of the first and second terms in Eqs. (53) and (54) is demonstrated. This equality is used in Sect. 3.3. An analogous demonstration can be carried out for rheological models with limited viscosity (Type A).

1.3 Derivation for the Maxwell model

The expression of G(t) for the Maxwell model is in the form of an exponential function as shown in Table 1. It is obvious that $ G(t{'} ) = G(t_{0}^{'} ) $ because $ t{'} $($ t{'} > t_{0} $) and $ t_{0}^{'} $ can be regarded as infinite. Substituting $ G(t{'} ) = G(t_{0}^{'} ) $ into Eqs. (55) and (57) yields $ F_{1}^{1} = F_{1}^{2} $. Therefore, the first terms from Eqs. (53) and (54) are equal. For any value of $ t{'} $ and $ t_{0}^{'} $, the second term of Eq. (53) can be written as:

$$ \begin{aligned} F_{2}^{1} &= 2\int_{0}^{{t^{'}_{{0}^{ - } }}} {\left[ {C_{1}^{\text{M}} \exp \left( { - \lambda_{{{\text{M}}2}} (t{'} - \tau )} \right)} \right]} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ & = \exp \left( { - \lambda_{{{\text{M}}2}} (t{'} - t_{0}^{'} )} \right) \times 2\int_{0}^{{t^{'}_{{0}^{ -} } }} {G(t_{0}^{'} - \tau )} \frac{{{\text{d}}e_{ij}^{\text{v}} }}{{{\text{d}}\tau }}{\text{d}}\tau \\ & = \exp \left( { - \lambda_{{{\text{M}}2}} (t^{'} - t_{0}^{'} )} \right)F_{2}^{2} . \\ \end{aligned} $$

(73)

The term $ \exp \left( { - \lambda_{{{\text{M}}2}} (t{'} - t_{0}^{'} )} \right) $ is non zero. The second term of Eq. (54) can be written as:

$$ F_{2}^{2} = 2\int_{0}^{{t{'}_{{0}^{ - }} }} {\left[ {C_{1}^{\text{M}} \exp \left( { - \lambda_{{{\text{M}}2}} (t_{0}^{'} - \tau )} \right)} \right]} \left[ {A_{ij}^{{4{\text{M}}}} + A_{ij}^{{5{\text{M}}}} \exp ( - \lambda_{{{\text{M}}1}} \tau )} \right]{\text{d}}\tau. $$

(74)

After integration and rearranging, yields:

$$ F_{2}^{2} = D_{ij}^{{1{\text{M}}}} \left[ {1 - \exp \left( { - \lambda_{{{\text{M}}2}} t_{0}^{'} } \right)} \right] + D_{ij}^{{2{\text{M}}}} \left[ {\exp \left( { - \lambda_{\text{M1}} t_{0}^{'} } \right) - \exp \left( { - \lambda_{{{\text{M}}2}} t_{0}^{'} } \right)} \right]. $$

(75)

When $ t{'} \to \infty $ and $ t_{0}^{'} \to \infty $, the above equation becomes:

$$ F_{2}^{2} = D_{ij}^{{1{\text{M}}}} \ne 0 $$

(76)

According to Eqs. (73) and (76), the second terms in Eqs. (53) and (54) are not equal, but hold the relationship shown in Eq. (73).

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Authors and Affiliations

H. N. Wang
- 1
- 2
Email author
S. Utili
- 3
M. J. Jiang
- 1
- 4
- 5
P. He
- 4

1.State Key Laboratory of Disaster Reduction in Civil EngineeringTongji UniversityShanghaiChina
2.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina
3.School of EngineeringUniversity of WarwickCoventryUK
4.Department of Geotechnical Engineering, College of Civil EngineeringTongji UniversityShanghaiChina
5.Key Laboratory of Geotechnical and Underground Engineering of Ministry of EducationTongji UniversityShanghaiChina

Analytical Solutions for Tunnels of Elliptical Cross-Section in Rheological Rock Accounting for Sequential Excavation

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