A hypo-plastic approach for evaluating railway ballast degradation

Abstract

Repetitive or cyclic rail loading deteriorates the engineering properties of the railway ballast by particle crushing and rearrangement. Most of the classical elasto-plastic models are unable to predict such ballast degradation despite successfully predicting the overall load–deformation behavior during cyclic densification. In this context, the present study delivers a novel hypo-plastic modeling approach coupled with breakage mechanics theory to bridge the gap of the conventional models. The hypo-plastic approach enables to predict the nonlinear load–deformation response of ballast-type granular materials for both monotonic and cyclic loading conditions, while circumventing the requirement of notional yield condition to predict the inelastic behavior. Breakage mechanics theory, on the other hand, establishes the links between particle comminution and the macroscopic deformation. The novelty of the proposed approach is threefold. Firstly, unlike the conventional hypo-plastic approaches, the development of the proposed model is within the continuum thermodynamics framework. Secondly, the model requires less number of physically identifiable parameters as compared to that of earlier models employed for assessing the particle breakage under cyclic loading. Third and finally, the numerical implementation of the model as a user-defined material is simple for solving boundary value problems. Under the compressive deformation regime, the model prediction of the ballast degradation along with the cyclic densification response agrees reasonably well with the experimental results found in the literature.

Appendices

Appendix 1: Tangent stiffness tensor

The incremental stress–strain relationship is

$$\dot{\sigma }_{ij} = (1 - \theta B)D_{ijkl} (\dot{\varepsilon }_{kl} - \dot{\varepsilon }_{kl}^{\text{p}} ) - \theta \frac{\sigma }{(1 - \theta B)}\dot{B}.$$

(25)

The consistency condition of the yield surface is given by:

$$\dot{y} = \frac{\partial y}{{\partial \sigma_{ij} }}\dot{\sigma }_{ij} + \frac{\partial y}{\partial B}\dot{B} = 0.$$

(26)

The nonnegative multiplier can be derived by substituting the stress increment (Eq. (25)) and flow rules (Eqs. (7–9)) in Eq. (26)

$$ \dot{\lambda} = \frac{{\frac{{\partial y}}{{\partial \sigma _{{ij}} }}\left( {1 - \theta B} \right)D_{{ijkl}} \dot{\varepsilon }_{{kl}}^{{}} }}{{\frac{{\partial y}}{{\partial \sigma _{{pq}} }}\left( {\left( {1 - \theta B} \right)D_{{pqrs}} \frac{{\partial y*}}{{\partial \sigma _{{rs}} }} + \frac{{\theta \sigma _{{pq}} }}{{\left( {1 - \theta B} \right)}}\frac{{\partial y*}}{{\partial E_{B} }}} \right) - \frac{{\partial y}}{{\partial B}}\frac{{\partial y*}}{{\partial E_{B} }}}}.$$

(27)

Finally, substituting the flow rules (Eqs. (7–9)) and the nonnegative multiplier (Eq. (27)) in the incremental stress–strain relation (Eq. (25)), the tangent stiffness tensor is derived.

$$L_{{ijkl}} ~ = \left( {1 - \theta B} \right)\left( {D_{{ijkl}} - \frac{{D_{{ijmn}} \left( {\left( {1 - \theta B} \right)\frac{{\partial y*}}{{\partial \sigma _{{mn}} }} - \theta \varepsilon _{{mn}}^{e} \frac{{\partial y*}}{{\partial E_{B} }}} \right)\frac{{\partial y}}{{\partial \sigma _{{st}} }}D_{{stkl}} }}{{\frac{{\partial y}}{{\partial \sigma _{{pq}} }}\left( {\left( {1 - \theta B} \right)D_{{pqrs}} \frac{{\partial y*}}{{\partial \sigma _{{rs}} }} + \frac{{\theta \sigma _{{pq}} }}{{\left( {1 - \theta B} \right)}}\frac{{\partial y*}}{{\partial E_{B} }}} \right) - \frac{{\partial y}}{{\partial B}}\frac{{\partial y*}}{{\partial E_{B} }}}}} \right),$$

(28)

where $\dot{\sigma }_{ij} = L_{ijkl} \dot{\varepsilon }_{kl}^{{}}$.

Appendix 2: Stress integration for hypo-plastic model

An explicit stress integration algorithm based on the multivariable Runge–Kutta method of order four (RK4) is used in the present study. Stress and breakage increment can be expressed in the following generalized form:

$$\dot{\sigma }_{ij} = k\left( {\sigma_{ij} , B} \right)\dot{\varepsilon }_{ij} ,$$

(29)

$$\dot{B} = j\left( {\sigma_{ij} , B} \right)\dot{\varepsilon }_{ij} .$$

(30)

According to RK4 method, at n^th step, the updated stress and breakage are given by,

$$\left. {\sigma_{ij} } \right|_{n + 1} = \left. {\sigma_{ij} } \right|_{n} + \frac{{\dot{\varepsilon }_{ij} }}{6}\left( {k_{ijkl}^{1} + 2k_{ijkl}^{2} + 2k_{ijkl}^{3} + k_{ijkl}^{4} } \right),$$

(31)

$$B_{n + 1} = B_{n} + \frac{{\dot{\varepsilon }_{ij} }}{6}\left( {j_{ij}^{1} + 2j_{ij}^{2} + 2j_{ij}^{3} + j_{ij}^{4} } \right).$$

(32)

In Eqs. (31, 32), ks and js are the fourth- and second-order tensors, respectively. $k_{ijkl}^{1}$ takes the similar form as tangent stiffness tensor (L _ijkl ), since $k_{ijkl}^{1}$ represents the slope between stress and strain,

$$k_{{ijkl}}^{1} = \left( {1 - \theta B} \right)\left( {D_{{ijkl}} - \frac{{D_{{ijmn}} \left( {\left( {1 - \theta B} \right)\frac{{\partial y*}}{{\partial \sigma _{{mn}} }} - \theta \varepsilon _{{mn}}^{e} \frac{{\partial y*}}{{\partial E_{B} }}} \right)\frac{{\partial y}}{{\partial \sigma _{{st}} }}D_{{stkl}} }}{{\frac{{\partial y}}{{\partial \sigma _{{pq}} }}\left( {\left( {1 - \theta B} \right)D_{{pqrs}} \frac{{\partial y*}}{{\partial \sigma _{{rs}} }} + \frac{{\theta \sigma _{{pq}} }}{{\left( {1 - \theta B} \right)}}\frac{{\partial y*}}{{\partial E_{B} }}} \right) - \frac{{\partial y}}{{\partial B}}\frac{{\partial y*}}{{\partial E_{B} }}}}} \right).$$

(33)

On the other hand, $j_{ij}^{1}$ is the slope of breakage evolution against strain, $\dot{B} = j_{ij} \dot{\varepsilon }_{ij}$. Therefore, substituting the nonnegative multiplier ($\dot{\lambda }$, Eq. (27)) in the breakage evolution law (Eq. (7)) one can find,

$$j_{{ij}}^{1} = \frac{{\frac{{\partial y}}{{\partial \sigma _{{ij}} }}\left( {1 - \theta B} \right)D_{{ijkl}} \frac{{\partial y*}}{{\partial E_{B} }}}}{{\frac{{\partial y}}{{\partial \sigma _{{pq}} }}D_{{pqrs}} \left( {\left( {1 - \theta B} \right)\frac{{\partial y*}}{{\partial \sigma _{{rs}} }} + \theta \varepsilon _{{rs}}^{e} \frac{{\partial y*}}{{\partial E_{B} }}} \right) - \frac{{\partial y}}{{\partial B}}\frac{{\partial y*}}{{\partial E_{B} }}}}.$$

(34)

The remaining, ks and js can be obtained from regular RK4 formulation

$$k_{ijkl}^{2} = k\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{1} \frac{{\dot{\varepsilon }_{kl} }}{2}, B_{n} + j_{kl}^{1} \frac{{\dot{\varepsilon }_{kl} }}{2}} \right),$$

(35)

$$k_{ijkl}^{3} = k\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{2} \frac{{\dot{\varepsilon }_{kl} }}{2},B_{n} + j_{kl}^{2} \frac{{\dot{\varepsilon }_{kl} }}{2}} \right),$$

(36)

$$k_{ijkl}^{4} = k\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{3} \dot{\varepsilon }_{kl} , B_{n} + j_{kl}^{3} \dot{\varepsilon }_{kl} } \right),$$

(37)

$$j_{kl}^{2} = j\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{1} \frac{{\dot{\varepsilon }_{kl} }}{2}, B_{n} + j_{kl}^{1} \frac{{\dot{\varepsilon }_{kl} }}{2}} \right),$$

(38)

$$j_{kl}^{3} = j\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{2} \frac{{\dot{\varepsilon }_{kl} }}{2},B_{n} + j_{kl}^{2} \frac{{\dot{\varepsilon }_{kl} }}{2}} \right),$$

(39)

$$j_{kl}^{4} = j\left( {\left. {\sigma_{ij} } \right|_{n} + k_{ijkl}^{3} \dot{\varepsilon }_{kl} , B_{n} + j_{kl}^{3} \dot{\varepsilon }_{kl} } \right).$$

(40)

The performance of the algorithm is evaluated by comparing the mechanical response of the hypo-plastic model for drained triaxial compression obtained using the forward Euler method. Figure 12 indicates better convergence in the stress–strain response obtained by using the RK4 method with higher strain increment.

Fig. 12

Stress–strain plot at different stress increment by using: a Euler forward method; b fourth-order Runge–Kutta method for drained triaxial compression