Creep characteristics and time-dependent creep model of tunnel lining structure concrete

Abstract

A creep test of concrete with different moisture contents is conducted using the MTS815.02 test system to investigate the creep damage law of concrete with respect to stress and time. The creep properties of concrete under hydration and the relationship between water content and concrete creep deformation are analyzed. The damage due to hydration and stress is considered comprehensively. The concrete creep characteristics and time-dependent creep model of different water contents are established based on rheological basic components. The Levenberg–Marquardt algorithm is used to fit the test data of concrete with different water contents. Furthermore, the correctness and rationality of the creep model are verified. Under the same stress level, the creep curves of different water contents show different creep stages. As the water content increases, creep changes at the same stress level are prone to stable creep and accelerated creep stages. The damage inside the concrete under high water content is significant, and the concrete is susceptible to creep deformation. The high degree of fit between the test data and the model curve reflects the suitability and feasibility of the established creep model to describe the deformation process of concrete creep with different water contents. The model accurately reflects not only the creep characteristics of the attenuation and stable creep stages but also overcomes the shortcomings of the traditional Nishihara model, which has difficulty in describing the accelerated creep. The deterioration law of creep parameters also reflects the damage degree of concrete under different water content conditions to some extent.

Appendices

Appendix A: Viscoelastic strain

The solution process in Eq. (15) is presented as follows. In the one-dimensional stress state, the viscoelastic rheological equation of concrete can be expressed as

$$ \sigma = E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} \varepsilon _{ve} + \eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} \varepsilon '_{ve}. $$

(30)

Equation (30) is transformed to obtain the following equation:

$$ \varepsilon _{ve} + \frac{\eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }\varepsilon '_{ve} = \frac{\sigma }{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }. $$

(31)

This first-order differential equation is nonhomogeneous and linear. Therefore, its general solution must be determined first. To determine its general solution, Eq. (31) can be transformed into the following form:

$$ \varepsilon _{ve} + \frac{\eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }\varepsilon '_{ve} = 0. $$

(32)

Eq. (32) is solved, and the general solution can be obtained as follows:

$$ \varepsilon _{ve} = C_{0}\exp \left \langle - \frac{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }{\eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }t \right \rangle = C_{0}\exp \left ( - \frac{a}{b}t \right ), $$

(33)

where $C_{0}$ is the integration constant, and $a$ and $b$ are the variable parameters.

The variable parameters $a$ and $b$ can be expressed as

$$ a = E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} ,\qquad b = \eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} . $$

(34)

According to Eq. (34), the following viscoelastic strain of concrete can be assumed to be expressed as

$$ \varepsilon _{ve} = \gamma \exp \left ( - \frac{a}{b}t \right ), $$

(35)

where $\gamma$ is the solution variable of the nonhomogeneous differential equation, which is also a function of time.

Equation (35) is subjected to the first-order partial derivative at time $t$, and the viscoelastic strain rate is obtained as follows:

$$ \varepsilon '_{ve} = \gamma '\exp \left ( - \frac{a}{b}t \right ) - \frac{b}{a}\gamma \exp \left ( - \frac{a}{b}t \right ). $$

(36)

Eqs. (35) and (36) are substituted into Eq. (31) and the following equation is obtained:

$$ \gamma \exp \left ( - \frac{a}{b}t \right ) + \frac{a}{b}\left [ \gamma '\exp \left ( - \frac{a}{b}t \right ) - \frac{b}{a}\gamma \exp \left ( - \frac{a}{b}t \right ) \right ] = \frac{\sigma }{a}. $$

(37)

The variable parameter $\gamma$ can be expressed as

$$ \gamma = \frac{\sigma }{a}\exp \left ( \frac{a}{b}t \right ) + C_{1}, $$

(38)

where $C_{1}$ is the integration constant.

Equation (38) is substituted into Eq. (35), and the viscoelastic strain of concrete is obtained as follows:

$$ \varepsilon _{ve} = \left [ \frac{\sigma }{a}\exp \left ( \frac{a}{b}t \right ) + C_{1} \right ]\exp \left ( - \frac{a}{b}t \right ) = \frac{\sigma }{a} + C_{1}\exp \left ( - \frac{a}{b}t \right ). $$

(39)

When $t = 0$, $\varepsilon _{ve} = 0$. The integration parameter $C_{1}$ is expressed as

$$ C_{1} = - \frac{\sigma }{a}. $$

(40)

In summary, the viscoelastic strain of concrete is expressed as

$$ \varepsilon _{ve} = \frac{\sigma }{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }\left \{ 1 \!-\! \exp \left [ - \frac{E_{1}\exp \left \{ - \left [ \alpha _{1}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }{\eta _{1}\exp \left \{ - \left [ \alpha _{2}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }t \right ] \right \} . $$

(41)

Appendix B: Viscoplastic strain

The viscoplastic rheological equation of concrete is expressed as

$$ \sigma = \sigma _{s} + \eta _{2}\exp \left \{ - \left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} \dot{\varepsilon }_{vp}. $$

(42)

Eq. (42) can be transformed into the following form:

$$ \dot{\varepsilon }_{vp} = \frac{\sigma - \sigma _{s}}{\eta _{2}\exp \left \{ - \left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} } = \frac{\sigma - \sigma _{s}}{\eta _{2}}\exp \left \{ - \left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} . $$

(43)

Eq. (43) is integrated to obtain the following viscoplastic strain:

$$ \varepsilon _{vp} = \frac{\sigma - \sigma _{s}}{\eta _{2}\left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]}\exp \left \{ \left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} + C_{2}t, $$

(44)

where $C_{2}$ is the integration constant.

When $t = 0$, $\varepsilon _{vp} = 0$. The integration parameter $C_{2}$ is expressed as

$$ C_{2} = - \frac{\sigma - \sigma _{s}}{\eta _{2}\left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t}. $$

(45)

Eq. (45) is substituted into Eq. (44) to obtain the following viscoplastic strain of concrete:

$$ \varepsilon _{vp} = \frac{\sigma - \sigma _{s}}{\left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t}\left \langle 1 - \frac{\exp \left \{ \left [ \alpha _{3}\left ( \sigma - \sigma _{A} \right ) + \beta \omega \right ]t \right \} }{\eta _{2}} \right \rangle . $$

(46)